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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/2017/paper-1/6/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 / 2017 / Paper 1 / 6 / b / Solution by

Codex  0 2026-10-18

The function graph is a horizontally shifted and scaled fourth power. It is nonnegative and touches the horizontal axis at its fourfold polynomial root. The axis intersections are

(- 1/2, 0) and (0, 1) .

(1)

The derivative is

8 (1 + 2 x)^{3}

, so the graph decreases to its flat local minimum at

x = - 1/2

and then increases. Its endpoint values on the requested interval are

f (- 1) = 1

and

f (1) = 81

. The original sketch includes a closer view of the minimum so that the small values are visible beside the much larger right endpoint.

Figure 1.
The fourth-power polynomial on the requested interval, with a close view of its flat minimum
. **Details**.

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