 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2019/paper-2/12/a/solution

The gradient vanishes when

f_{x} = x (1 + 2 y) = 0, f_{y} = y + x^{2} - y^{2} = 0.

x = 0

y = 0

y = 1

. If

y = - 1/2

, it gives

x^{2} = 3/4

. These exhaust the possibilities. The stationary points and their values are

(x, y) (0, 0) (0, 1) (3 /2, - 1/2) (- 3 /2, - 1/2) f (x, y) 0 1/6 1/6 1/6 .

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