 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-2/3/a/solution

f^{'} (x) = 1/ x

. The chord slope is

(lo g 2 - lo g 1) / (2 - 1) = lo g 2

. Hence the mean value theorem point satisfies

1/ ξ = lo g 2

ξ = \frac{1}{lo g 2} .

It lies strictly between one and two. For example,

lo g 2 = \int_{1}^{2} d u / u

lies strictly between

1/2

and one.

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