1. What is the maximum of f(x,y) = x²y, given x² + y² = 1 ?

Since,

Substituting into the given function f(x,y), yields,

By Fermat’s theorem, if f has a local extremum, that is, a maximum or minimum value at c, and if f’(c) exists, then f’(c) = 0.

So, take the derivative with respect to y of equation (1) and set it equal to zero to find the value that will give us the maximum value of f(x,y).

Since (d/dy)f(x,y) goes from a positive to a negative value at the point 3^1/2/3, the maximum value of f(x,y) can be obtained by choosing y = 3^1/2/3 and substituting y into equation (1) as follows,

Notice that choosing y = -3^1/2/3 yields the minimum value of f(x,y).