Find the maximum and minimum values of the function f(x, y)=2x2+3y2−4x−5 on the domain x2+y2≤100. The maximum value of f(x, y) is:

First find the critical points of f :

so the point (1, 0) is the only critical point, at which we have

Next check for critical points along the boundary, which can be found by converting to polar coordinates:

Find the critical points of g :

where n is any integer. We get 4 critical points in the interval [0, 2π) at

So f has a minimum of -7 and a maximum of 299.

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step-by-step explanation: