 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/20/a/solution

Let

G = x^{2} / a^{2} + y^{2} / a^{2} + z^{2} / b^{2} - 1

. Its gradient cannot vanish on the ellipsoid, since the origin is not on the surface. Thus the Lagrange multiplier equations capture every constrained stationary point:

4 x^{3} = \frac{2 λ x}{a ^{2}}, 2 y z^{2} = \frac{2 λ y}{a ^{2}}, 2 y^{2} z = \frac{2 λ z}{b ^{2}}, \frac{x ^{2} + y ^{2}}{a ^{2}} + \frac{z ^{2}}{b ^{2}} = 1.

T \geq 0

. Equality requires

x = 0

and

yz = 0

. The surface constraint then gives the four coldest points:

T_{m i n} = 0, (x, y, z) = (0, \pm a, 0) or (0, 0, \pm b) .

These are all such points, because a sum of nonnegative terms can vanish only when each term vanishes.

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