DIFFERENTIATION AND LOCAL EXTREMA OF MULTIVARIATE FUNCTIONS

Functions of two variables take two real numbers and assign a third real number to them. In other words, they assign a third number to a pair of numbers.

We could look at these number-pairs as coordinates in a plane.

Functions of two variables assign a third coordinate, the height, to the points of this plane.

By assigning this third (height) coordinate to all points of the domain, a surface is taking shape above the x,y plane. This is the graph of the function.

Some properties of single variable functions can be transmitted to two-variable functions, but some properties cannot.

There is no point for instance talking about monotonicity in case of two-variable functions, as it would be quite difficult to determine whether a surface happens to be increasing or decreasing.

On the other hand, the concept of minimum and maximum can be transmitted.

We should imagine the maximum of a two-variable function as a peak of a mountain,

and the minimum as a valley.

Let's see some two-variable functions.

LOCAL MINIMUM

SADDLE POINT

LOCAL MAXIMUM

Our task is to find out where the minimum,

the maximum or even the saddle point of a two-variable function happens to be.

Just like in the one-variable case, we will have to differentiate here, too, but now we have x as well as y, so we have to differentiate with respect to x and also with respect to y, which should be twice as much fun.

These derivatives are called partial derivatives.

Let's see the partial derivatives.

Let’s differentiate this function, for instance.

PARTIAL DERIVATIVES

PARTIAL DERIVATIVE OF FUNCTION WITH RESPECT TO

we differentiate with respect to x, while y is held constant

differentiate with respect to x

y is treated as a constant,

if it stands by itself, its derivative is zero

if it is multiplied by some expression with x, than it stays as is

PARTIAL DERIVATIVE OF FUNCTION WITH RESPECT TO

we differentiate with respect to y, while x is held constant

differentiate with respect to y

x is treated as a constant,

if it stands by itself, its derivative is zero

if it is multiplied by some expression with y, than it stays as is.

There is another notation for partial derivatives.

This

We will use both notations.

Here comes another function, let’s differentiate this one, too.

SECOND ORDER DERIVATIVES

Both first order partial derivatives can be further differentiated with respect to x as well as y, too.

This way we get four second order derivatives.

The two outer ones are called pure second order derivatives,

and the two middle ones are the mixed second order derivatives.

The two mixed second order derivatives are usually equal.

Well, to be exact, they are equal if the function is twice totally differentiable.

But instead, we should remember that they are always equal, except in the section that is for professionals only, where the precise definitions of multivariate differentiation will be discussed.

Now, let’s see how we can find local minima and maxima using partial differentiation.