WBJEE 2012 Math Solved Paper

Given, f(x)‌f′(x)<0
⇒ f(x) and f'(x) must be of opposite sign
(i) Let f(x)=e−x
∴f(x)=−e−x
⇒f(x)>0 and f′(x)<0,∀x∊R
(ii) Let f(x)=−e−x
∴f′(x)=e−x
⇒ f(x)<0 and f′(x)>0,∀x∊R
But |f(x)|=|±e−x|=e−x in both cases
∴

d

dx

|f(x)|=−e−x<0 in both cases,∀ x ∊ R
⇒|f(x)| must be a decreasing function