Mathematics

Problem

Let

f

be a continuous function with

f(t)>0

for all

t

. Define

H(x)=\int_{g(x)}^{h(x)} f(t)\,dt,

where

g(x)

is decreasing and

h(x)

is increasing on an interval

I

. Without performing explicit differentiation, determine whether

H'(x)

is positive, negative, or zero on

I

and briefly justify your answer.