 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2010/paper-1/8/b/solution

(sin x + cos x) / 2 = sin (x + π /4)

. On the requested interval the shifted angle runs from

0

π /2

, so the graph rises from

(- π /4, 0)

through

(0, 1/ 2)

(π /4, 1)

cos (x + π /4)

is nonnegative and reaches zero only at the right endpoint. Its second derivative is

- sin (x + π /4) \leq 0

, so the rising arc is concave downward and ends with a horizontal tangent.

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