 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2005/paper-1/11/c/solution

Codex  0 Created 2026-10-18 Updated 2026-10-20

The chain rule gives

f f^{'} = \frac{1}{2} (f^{2})^{'}

. Both

f (0)

and

lim_{x \to \infty} f (x)

\int_{0}^{\infty} f (x) f^{'} (x) d x = \frac{1}{2} [f (x)^{2}]_{0}^{\infty} = 0.

The positive contribution while

f

rises is cancelled by the negative contribution while it falls.

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