Local minimum, local maximum, increase, decrease, Convex, Concave, Domain, x-intercept, First derivative, Monotonicity, Second derivative, Concavity, Limits, Range, Shape of the function.
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 is anything
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
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.
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.