Question

Let $f\left(x\right)$ and $g\left(x\right)$ are differentiable function such that $g\left(x\right)=2xf\left(x\right)+x^2{f}^{\prime }\left(x\right)$ in $[a,d]$ and $0<a<b<c<d,f\left(a\right)=0,f\left(b\right)=5,f\left(c\right)=-3,f\left(d\right)=0,$ then the minimum number of zero(s) for $g\left(x\right)=0$ is

Answer 1

Given : $g\left(x\right)=2xf\left(x\right)+x^2{f}^{\prime }\left(x\right)=\frac{d\left(x^{2}f\right(x\left)\right)}{dx}$
also $f\left(x\right)$ changes its sign for $(b,c),$ so there is atleast one root of $f\left(x\right)=0$ in $(b,c)$ and $f\left(a\right)=f\left(d\right)=0,$
So, $f\left(x\right)=0$ have atleast $3$ roots.
$\Rightarrow h\left(x\right)=x^{2}f\left(x\right)$ has one more root i.e. $x=0,$ which does not lie in $[a,d]$
$\Rightarrow g\left(x\right)=h^{\prime }\left(x\right)$ will have atleast $2$ roots.