 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/b/solution

H (x, y) = (1 + 2 y 2 x 2 x 1 - 2 y) .

At the origin it is the identity, so the origin is a strict local minimum. At

(0, 1)

it is

diag (3, - 1)

, so this point is a saddle point. At either remaining stationary point,

H = (0 \pm 3 \pm 3 2), det H = - 3 < 0,

which again gives a saddle point. All three saddle points have eigenvalues

3

and

- 1

; there are no local maxima.

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