 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/a/solution

Differentiating gives

y^{'} = e^{s i n^{2} x} sin 2 x, y^{''} = e^{s i n^{2} x} (sin^{2} 2 x + 2 cos 2 x) .

The exponential never vanishes, so all stationary points have

x = kπ /2

. At

x = kπ

y = 1

and

y^{''} = 2 > 0

; at

x = π /2 + kπ

y = e

and

y^{''} = - 2 e < 0

. Thus

(kπ, 1) are minima, (π /2 + kπ, e) are maxima, k \in Z .

These are global minima and maxima because

0 \leq sin^{2} x \leq 1

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