Mathematics

Mathematics

Problem

A continuous function

f

satisfies

f\bigl(\tfrac12\bigr)<0

and

f(0)>0

. Define

h(x)=\displaystyle\int_{0}^{\cos x} f(t)\,dt

. Which statement best describes the behavior of

h

at

x=\tfrac{\pi}{3}

? a)

h

is decreasing at

x=\tfrac{\pi}{3}

because

h'(\tfrac{\pi}{3})<0

. b)

h

has a relative maximum at

x=\tfrac{\pi}{3}

because

h'(\tfrac{\pi}{3})=0

. c)

h

is increasing at

x=\tfrac{\pi}{3}

because

h'(\tfrac{\pi}{3})>0

. d)

h

has a relative maximum at

x=\tfrac{\pi}{3}

because

h'(\tfrac{\pi}{3})>0

.

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