A continuous function satisfies

MATHEMATICS · Single correct · CBSE · Mathematics · Integrals · Integrals

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

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