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