# Graph analysis, optimization problems

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease, marital status and criminal record.

Well, while some of these questions can be answered by the FBI only, there are some, for instance increase and decrease that we could identify fairly easily.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

But we not only want to know when the function increases and decreases, but also how it does those things.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

There is no radical or logarithm this time, but there is a fraction. So, the denominator cannot be zero: , and this can also be written as:

If time allows, it may be worth to find the x-intercept of the function, which is the solution of the

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

The steps will be exactly the same as in the previous slides, but there will be some surprises on the way.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease, marital status and criminal record.

Well, while some of these questions can be answered by the FBI only, there are some, for instance increase and decrease that we could identify fairly easily.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

But we not only want to know when the function increases and decreases, but also how it does those things.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

There is no radical or logarithm this time, but there is a fraction. So, the denominator cannot be zero: , and this can also be written as:

If time allows, it may be worth to find the x-intercept of the function, which is the solution of the

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

The steps will be exactly the same as in the previous slides, but there will be some surprises on the way.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Changes in the current price of a stock between 8:00am and 6:00pm are described by the following function, where at the xth hour of the day the stock price in thousand dollars is:

What was the opening and the closing price?

At which hour was the price the lowest and the highest?

At opening x=8, and at closing x=18, so we need to substitute these.

Now let’s see when the price was lowest and highest.

We need to differentiate for this:

Well, it appears that the maximum was at 8:00am, and the minimum was at noon.

The demand function of a product is:

where x denotes the unit price of the product. What unit price would ensure the highest income?

We have to differentiate again, but not this function, because this is the demand.

We need the revenues function.

Let’s see what the revenues is.

We have to multiply the quantity sold...

by the unit price.

Splendid.

And now we can differentiate.

The maximum revenue is at unit price 10.

Now, that we know all this, let’s try one last problem.

The per-unit profit of a product in dollars is:

where x is the weekly sales quantity, in 1000 units. What sales volume would result in optimal total weekly profit?

The per-unit profit means the amount of profit we make when we sell one product unit. Let’s say the per-unit profit is $5, and we sell 100 units. Then the total profit is $500.

The total profit is computed by multiplying the per-unit profit by the volume sold.

quantity

Let's see what the volume is.

Well, it is x, but given in 1000 units.

So, x=2 does not mean they sell 2 units, it means they sell 2000 units. If x=3, then they sell 3000.

So, the quantity sold is 1000x.

Splendid.

And now we can differentiate.

The maximum is at x=1, in other words, the weekly profit is maximized when 1000 units are sold.

FV04

We are left with x=0

FV06

Fractions always have to be changed to a common denominator before testing for signs!

It seems we are out of luck: there is no x-intercept.

FV07

We are not going to bother with this, it doesn’t worth it, let’s differentiate instead!

Due to x>3, we say good bye to this.

FV08

Never zero, it is negative for all x values

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease, marital status and criminal record.

Well, while some of these questions can be answered by the FBI only, there are some, for instance increase and decrease that we could identify fairly easily.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

But we not only want to know when the function increases and decreases, but also how it does those things.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

There is no radical or logarithm this time, but there is a fraction. So, the denominator cannot be zero: , and this can also be written as:

If time allows, it may be worth to find the x-intercept of the function, which is the solution of the

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

The steps will be exactly the same as in the previous slides, but there will be some surprises on the way.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.

Here is this function

and our task is to analyze the function’s behavior.

We are mostly interested in features such as locations of minima, maxima, increase, decrease.

We can analyze these behaviors of the function by using the miracle of differentiation.

We can identify increase and decrease by using differentiation.

If the derivative of the function is positive, then the function increases,

if the derivative is negative, then the function decreases.

We call the function concave on those sections where it is in a sad mood like this,

and we call it convex, where it is happy.

The mood of the function is indicated by its second derivative.

If the second derivative is negative, then the function is concave,

if positive, then it is convex.

The story starts with determining the x values where the function is actually defined.

It is worth to remember a few things:

this here

this is anything

fraction’s denominator

If we don’t bump into any of these, then the function is defined for all x values.

(Except when there are tangents and other horrors.)

equation. This time it is

eANYTHING>0

STEP 2. Testing for monotonicity: differentiation

POSITIVE FOR SURE

NEGATIVE FOR SURE

And now we check when the derivative is positive and when negative.

Let’s show the signs of the numerator and the denominator separately on this number line.

If one of them is positive and the other is negative, then the fraction is negative.

If both are negative, then the fraction is positive.

And this final part is negative again.

Now we summarize these in a table.

monotonicity

convexity

Let’s push these over a little.

STEP 3. The sign of the derivative

STEP 4. Testing for concavity: second derivative

STEP 5. The sign of the second derivative

STEP 6. Limits at the endpoints of Df

STEP 7. Rf

And now let’s analyze another function.

But first things first.

If we recall, we have to watch for these four things:

If we don’t bump into any of these, then the domain consists of all real x values.

This time we don’t.

We draw the signs of the factors one by one onto a number line.

Negative is indicated by a dashed line, and positive by a solid line.

Well, it doesn’t hurt to know the limits of the ex function at negative and positive infinity.

Now, this limit will be funny.

Namely, while one factor of the product tends to infinity,

the other one tends to zero.

And is a critical limit.

The product can be zero, but it can also be infinity, and it can even be negative infinity.

For critical limits like this, to decide the result, we need to identify which one is stronger.

It is possible that the first one is the stronger one...

but it is also possible, that it is the other.

And now let’s see how strong they are.

This ranking tells us how fast each function tends to its own limit.

Applying this to our specific problem, well, it seems ex is stronger.