 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-2/18/c/solution

For the composite square root,

y > 1

throughout the restricted

x

interval. Differentiate with respect to

y

\frac{d z}{d y} = \frac{1}{2 y lo g y}, \frac{d ^{2} z}{d y ^{2}} = - \frac{2 lo g y + 1}{4 y ^{2} ( lo g y ) ^{3/2}} .

The second expression comes from differentiating both

y^{- 1}

and

(lo g y)^{- 1/2}

. The endpoints where

y = 1

are excluded here.

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