JEE Main 2014 (Online) 12th AprilQuestion 1
Let $f$ and $g$ be two differentiable functions on $R$ such that $f'(x)>0$ and $g'(x)<0$ for all $x\in R$. Then for all $x$ :
Differentiation and its Application
Solution
Concept. $g$ decreasing reverses an inequality; $f$ increasing keeps it. Formulas used. (1) $g'<0\Rightarrow g(x)>g(x+1)$; $f'>0\Rightarrow(a>b\Rightarrow f(a)>f(b))$Step 1 - Apply $g$. By (1), $g(x)>g(x+1)$. Step 2 - Apply $f$. $f(g(x))>f(g(x+1))$. Answer: (B)