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.

(1, –1), (16, –4), (36, –6)

step-by-step explanation:

given that the function is

this can also be written as

we have to graph this function and check which list of points lie on the graph

see attachment for graph:

we find that the points lying are (1,-1) (16,-4) and (36,-6)

since square root is there we find that x can only be posiitve but y can be since square root both positive and negative.