## 8.2 Optimizing F(x,y) holding G(x,y) Constant

We want to find extrema of a function F of two variables, x and y, subject
to the condition that some other function G, is constant.

A change in the variable x will usually require a corresponding change in y
for G to stay constant.

Let
be the derivative of F with respect to x if y does not change at all, and
the derivative of F with respect to y if x does not change.

These are called **partial derivatives**. If both x and y change infinitesimally,
then we have
,and similarly
.

The condition that G not change is that dG = 0; the condition that F be critical
is that dF = 0.

We get

and we can eliminate dx and dy, getting the condition:

whose solution gives us the desired extrema.

Tin can example

Another example