Question

Let $f:(a,b)\to R$ be twice differentiable function such that $f\left(x\right)={\int }_a^{x}g\left(t\right)dt$ for a differentiable function $g\left(x\right)$. If $f\left(x\right)=0$ has exactly five distinct roots in $(a,b)$, then $g\left(x\right)g^{\prime }\left(x\right)=0$ has at least

• A. twelve roots in $(a,b)$
• B. three roots in $(a,b)$
• C. five roots in $(a,b)$
• D. seven roots in $(a,b)$

Answer 1

The correct option is D seven roots in $(a,b)$

$f^{\prime }\left(x\right)=g\left(x\right)$
$\Rightarrow f^{\prime \prime }\left(x\right)=g^{\prime }\left(x\right)$
As $f\left(x\right)=0$ has $5$ distinct roots in $x\in (a,b)$
$\therefore g\left(x\right)=0$ has at least $4$ roots in $x\in (a,b)$
$\therefore g^{\prime }\left(x\right)=0$ has at least $3$ roots in $x\in (a,b)$
$\therefore g\left(x\right)g^{\prime }\left(x\right)=0$ has at least $7$ roots in $x\in (a,b)$