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ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2017/prelims/mathematics-iii/3/c/solution

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Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics III / 3 / c / Solution by

Codex  0 2026-10-25

Let

F = x + y + z^{2}

and

g = x^{2} + y^{2} + z^{2}

. The unit sphere is compact, so the continuous function

F

has a global maximum and global minimum. Since

\nabla g \neq = 0

on the unit sphere, all such points satisfy the Lagrange multiplier equations

1 = 2 λ x, 1 = 2 λ y, 2 z = 2 λ z, x^{2} + y^{2} + z^{2} = 1.

Thus

x = y

and either

z = 0

or

λ = 1

.

If

z = 0

, then

x = y = \pm 1/ 2

F = \pm 2

. If

λ = 1

, then

x = y = 1/2

and

z = \pm 1/ 2

F = 3/2

. These exhaust the candidates, so

min F = - 2 at (- 1/ 2, - 1/ 2, 0), max F = \frac{3}{2} at (1/2, 1/2, \pm 1/ 2) .

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