 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/16/a/solution

Translate the coordinates by

u = x - 1

and

v = y + 1

, and write

E = e^{- (u^{2} + v^{2}) /2} > 0

f_{x} = v (1 - u^{2}) E, f_{y} = u (1 - v^{2}) E .

u = 0

v = 0

, these equations together force

u = v = 0

. Otherwise they require

u^{2} = v^{2} = 1

. Thus the complete set of stationary points is

(x, y) = (1, - 1), (2, 0), (0, - 2), (2, - 2), (0, 0) .

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