 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2023/paper-1/16/b/solution

F (x) = ⎩ ⎨ ⎧ 0, 4 x^{3} - 3 x^{4}, 1, x < 0, 0 \leq x \leq 1, x > 1.

Thus the function graph is horizontal at zero for

- 1 \leq x \leq 0

, rises continuously from

(0, 0)

(1, 1)

, and is horizontal at one for

1 \leq x \leq 2

12 x^{2} (1 - x)

is positive on

(0, 1)

and zero at both endpoints. The second derivative

12 x (2 - 3 x)

changes sign at

x = 2/3

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