 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/32a/i/solution

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

For

x \in [0, 1]

, the nonnegative function

x (1 - x^{2})

1 - 3 x^{2}

, so its maximum occurs at

x = 1/ 3

and equals

\frac{1}{3} (1 - \frac{1}{3}) = \frac{2}{3 3} .

The map preserves the interval exactly when this maximum is at most one:

λ \frac{2}{3 3} \leq 1.

Thus the upper limit is

λ_{m a x} = \frac{3 3}{2} .

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