JEE Main · 2021 · 27 Jul 2021 · Shift 2 · Application of Derivatives

Differentiating $f(x)={\int }_a^{x}g(t)dt$ gives $f^{\prime }(x)=g(x)$ and $f^{\prime \prime }(x)=g^{\prime }(x)$. Between two consecutive roots of $f$, Rolle's theorem produces a root of $f^{\prime }=g$, so five roots of $f$ give at least four roots of $g$ in $(a,b)$. Applying Rolle's theorem again between consecutive roots of $g$ gives at least three roots of $g^{\prime }$, and each of these lies strictly between two roots of $g$, so it is distinct from all of them. The product $g(x)g^{\prime }(x)$ vanishes at every one of these points, so $g(x)g^{\prime }(x)=0$ has at least seven roots in $(a,b)$.