JEE Main 2020 (Online) 9th January Morning ShiftQuestion 1
Let $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$. If for all $x\in(a,b)$, $f'(x)>0$ and $f''(x)<0$, then for any $c\in(a,b)$, $\frac{f(c)-f(a)}{f(b)-f(c)}$ is greater than :
Differentiation and its Application
Solution
Concept. On a concave increasing curve, chord $AC$ is steeper than chord $CB$.Formulas used. (1) MVT: $\dfrac{f(c)-f(a)}{c-a}=f'(\xi_1)$, $\ \dfrac{f(b)-f(c)}{b-c}=f'(\xi_2)$, with $a<\xi_1<c<\xi_2<b$Step 1 - Compare slopes. $f''<0$, so $f'$ is decreasing, and $\xi_1<\xi_2$ gives $f'(\xi_1)>f'(\xi_2)$. By (1), $$\frac{f(c)-f(a)}{c-a}>\frac{f(b)-f(c)}{b-c}.$$ Step 2 - Rearrange. $f(b)-f(c)>0$ (as $f'>0$), so $$\frac{f(c)-f(a)}{f(b)-f(c)}>\frac{c-a}{b-c}.$$ Answer: (B)