JEE Main 2020 (Online) 6th September Evening ShiftQuestion 1
For all twice differentiable functions $f:\mathbf{R}\to\mathbf{R}$, with $f(0)=f(1)=f'(0)=0$,
Differentiation and its Application
Solution
Concept. Rolle on $f$ gives a zero of $f'$ inside $(0,1)$; together with $f'(0)=0$, Rolle on $f'$ gives a zero of $f''$. Formulas used. (1) Rolle: $g(p)=g(q)$ $\Rightarrow$ $g'(c)=0$ for some $c\in(p,q)$Step 1 - Zero of $f'$. $f(0)=f(1)$, so by (1) $f'(c)=0$ for some $c\in(0,1)$. Step 2 - Zero of $f''$. $f'(0)=f'(c)=0$, so by (1) $f''(d)=0$ for some $d\in(0,c)\subset(0,1)$. Step 3 - Others. (A) is false by Step 2; (B), (D) fail for e.g. $f=x^2(x-1)$. Answer: (C)