 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/12/c/solution

A stationary point must satisfy both factored equations from part (a). If either coordinate is zero, the other equation forces the other coordinate to vanish. This gives

(0, 0)

Otherwise

x^{4} = 8 y^{2}

and

y^{4} = 8 x^{2}

. With

u = x^{2} > 0

and

v = y^{2} > 0

, these become

u^{2} = 8 v

and

v^{2} = 8 u

. Dividing gives

(u / v)^{3} = 1

, so

u = v

. Then

u^{2} = 8 u

u = v = 8

. Hence all five stationary points are

(0, 0), (22 ε, 22 δ), ε, δ \in {- 1, 1} .

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