Mathematics - Application of Derivative Question with Solution | TestHub

$h^{\prime }(x)=f^{\prime }(x)-2f(x)f^{\prime }(x)+3(f(x){)}^2{f}^{\prime }(x)$

$=f^{\prime }(x)\left[1-2f(x)+3(f(x){)}^2\right]$

Since, $1-2\mathrm{f}(\mathrm{x})+3(\mathrm{f}(\mathrm{x}){)}^2>0$ for all $\mathrm{f}(\mathrm{x})$ $\Rightarrow h^{\prime }(x)>0\text{ if }{f}^{\prime }(x)>0$

$\Rightarrow h$ is increasing when ever $f$ is increasing and $h^{\prime }(x)<0$ if $f^{\prime }(x)<0$

$\Rightarrow \mathrm{h}$ is decreasing when ever f is decreasing.