ISI 2017 PEA

[chirag]

Let f : R → R be a differentiable function such that f(x)f'(x)<0
 for all x ∈ R. Then
A. f(x) is an increasing function B. |f(x)| is an increasing function C. f(x) is a
decreasing function D. |f(x)| is a decreasing function

[Nak]

The answer is |f(x)| is a decreasing function.

f(x)f'(x)< 0 means that both have opposite sign

First case.

f(x) is greater than 0.
So |f(x)| = f(x)

Differentiating you will get f'(x) which has to be less than 0 as they have opposite sign

Second case

f(x) < 0

|f(x)| = -f(x)

Differentiating you will get -f'(x)  since f(x) is less than 0, f'(x) will be greater than 0 and hence -f'(x) < 0

So in both cases f'(x) < 0 so it is a decreasing function

[chirag]

Thanks Nak.