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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2003/paper-1/3/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 3 / b / Solution by

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

The function is odd, with a horizontal tangent line at the origin and tails approaching zero from below on the negative side and above on the positive side. It decreases to its negative minimum at

x = - 1

, rises through the stationary inflection point at zero to its positive maximum at

x = 1

, then decreases toward zero.

Figure 1.

An odd cubic Gaussian profile with two extrema and a stationary inflection

. Original function plot with the three stationary points marked. The near-zero cubic behavior explains the flat crossing without an extremum. Details.

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