Consider the following equation. f(x, y) = e−(x − a)2 − (y − b)2 (a) find the critical points. (x, y) = a, b (b) find a and b such that the critical point is at (−3, 8). a = b = (c) for the values of a and b in part (b), is (−3, 8) a local maximum, local minimum, or a saddle point?

a.

Critical points occur where . The exponential factor is always positive, so we have

b. As the previous answer established, the critical point occurs at (-3, 8) if and .

c. Check the determinant of the Hessian matrix of :

The second-order derivatives are

so that the determinant of the Hessian is

The sign of the determinant is unchanged by the exponential term so we can ignore it. For and , the remaining factor in the determinant has a value of 4, which is positive. At this point we also have

which is negative, and this indicates that (-3, 8) is a local maximum.

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