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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