 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/e/solution

12 x (2 - 3 x) = 0

with

0 < x < 1

x = 2/3

. The sign change of

f^{'}

established in part (d), together with zero endpoint values, makes this the global maximum. Thus

mode = \frac{2}{3}, f (2/3) = \frac{16}{9} .

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