Let and be differentiable functions such that and for all

MATHEMATICS · Single correct · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

p(x)

and

q(x)

be differentiable functions such that

p(x)>0

and

q(x)>0

for all

x

. At

x=c

it is known that

p'(c)>0

and

q'(c)<0

. For the product

f(x)=p(x)q(x)

, which statement about

f'(c)

must be true? (a)

f'(c)>0

(b)

f'(c)=0

f'(c)

depends on the relative magnitudes of

p(c)

q(c)

p'(c)

, and

q'(c)

. (d)

f'(c)<0