Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Let, $g(x)=\frac{d}{dx}\left[f(x)\cdot f^{\prime }(x)\right]$ to get the zero of $g(x)$ we take function $h(x)=f(x)\cdot f^{\prime }(x)$ between any two roots of $h(x)$ there lies atleast one root of $h^{\prime }(x)=0$ $$\begin{array}{rl}& \Rightarrow g(x)=0\\ & \Rightarrow h(x)=0\\ & \Rightarrow f(x)=0\text{ or }{f}^{\prime }(x)=0\\ & \text{ If }f(x)=0\text{ has }4\text{ minimum solutions, }\\ & f^{\prime }(x)=0\text{ has }3\text{ minimum solutions, }\\ & h(x)=0\text{ has }7\text{ minimum solutions, then }\\ & h^{\prime }(x)=g(x)=0\text{ has }6\text{ minimum solutions. }\\ & \end{array}$$