Mathematics - Quadratic Equation Question with Solution | TestHub

$$g(x)=f^{\prime }(x{)}^2+f^{\prime \prime }(x)f(x)=\frac{d}{dx}[f(x)f^{\prime }(x)]$$

Let $h(x)=f(x)f^{\prime }(x)$.

Then, $f(x)=0$ has four roots, namely $a,b,c,e$,

where $b<c$ and $c<d$.

And $f^{\prime }(x)=0$ at three points $k_1,k_2,k_3$ where

$a<k_1<b$, $b<k_2<c$, $c<k_3<e$.

[Because between any two roots of a polynomial function $f(x)=0$, there lies at least one root of $f^{\prime }(x)=0$].

There are at least 7 roots of $f(x)f^{\prime }(x)=0$. There are at least 6 roots of $\frac{d}{dx}[f(x)f^{\prime }(x)]=0$, i.e., of $g(x)=0$.