 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2009/paper-2/13/a/i/solution

Put

X = x - 1

and

H = e^{- (X^{2} + y^{2}) /2} > 0

. The gradient components are

z_{x} = y (1 - X^{2}) H

and

z_{y} = X (1 - y^{2}) H

. If either

X

y

vanishes, both must vanish. Otherwise both squared coordinates must be one. Therefore the five stationary points in the original

(x, y)

coordinates are

(1, 0), (0, 1), (0, - 1), (2, 1), (2, - 1) .

 New to topics? Read the docs here!