JEE Main 2014 (Offline)Question 1
If $f$ and $g$ are differentiable functions in $[0,1]$ satisfying $f(0)=2=g(1)$, $g(0)=0$ and $f(1)=6$, then for some $c\in\,]0,1[$
Differentiation and its Application
Solution
Concept. Choose $k$ so that $f-kg$ has equal values at $0$ and $1$; then apply Rolle. Formulas used. (1) Rolle: $h(0)=h(1)\Rightarrow h'(c)=0$ for some $c\in(0,1)$Step 1 - Choose $h$. $h=f-2g$: $h(0)=2-0=2$ and $h(1)=6-4=2$. Step 2 - Apply (1). $h'(c)=f'(c)-2g'(c)=0$, so $f'(c)=2g'(c)$. Answer: (B)