JEE Main 2021 (Online) 27th July Evening ShiftQuestion 1
Let $f:(a,b)\to\mathbf{R}$ be twice differentiable function such that $f(x)=\int_a^xg(t)\,dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct roots in $(a,b)$, then $g(x)g'(x)=0$ has at least :
Differentiation and its Application
Solution
Concept. $f'=g$. Rolle on $f$ gives roots of $g$; Rolle on $g$ gives roots of $g'$, lying strictly between roots of $g$. Formulas used. (1) Rolle: between two zeros of a differentiable function there is a zero of its derivativeStep 1 - Roots of $g$. $f$ has $5$ distinct roots. By (1), $g=f'$ has at least $4$, one in each gap. Step 2 - Roots of $g'$. By (1) on $g$, $g'$ has at least $3$, each strictly between two roots of $g$, so different from them. Step 3 - Count. $g\,g'=0$ has at least $4+3=7$ roots. Answer: (C)