Let (x, y) = (x2 + y2)e x2y2 . (a) Where does f take on its minimum value? Do not use calculus to answer this

Let ƒ(x, y) = (x2 + y2)e^−x2−y2.

(a) Where does f take on its minimum value? Do not use calculus to answer this question. (b) Verify that the set of critical points of f consists of the origin (0,0) and the unit circle x + y = 1. (c) The Second Derivative Test fails for points on the unit circle (this can be checked by some lengthy algebra). Prove, however, that f takes on its maximum value on the unit circle by analyzing the function g(t)= te for t > 0.