 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-3/2f/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

F (x, y, z) = x^{2} + y^{2} + z^{3} + a z + b .

F^{- 1} (0)

\nabla F \neq = 0

. A critical point on the level set would satisfy

x = y = 0, a = - 3 z^{2}, b = 2 z^{3} .

These equations imply

4 a^{3} + 27 b^{2} = 0

. Therefore

4 a^{3} + 27 b^{2} \neq = 0 ⟹ S_{a, b} is a smooth surface .

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