Question

Let $\mathrm{h}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)-(\mathrm{f}{\left(\mathrm{x}\right))}^2+(\mathrm{f}{\left(\mathrm{x}\right))}^3$ for every real number $\mathrm{x}$. Then :

• A. $\mathrm{h}$ is increasing whenever $\mathrm{f}$ is increasing
• B. $\mathrm{h}$ is increasing whenever $\mathrm{f}$ is decreasing
• C. $\mathrm{h}$ is decreasing whenever $\mathrm{f}$ is increasing
• D. nothing can be said in general

Answer 1

The correct option is A

$\mathrm{h}$ is increasing whenever $\mathrm{f}$ is increasing

Determining the correct option

We have the equation : $\mathrm{h}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}\right)-(\mathrm{f}{\left(\mathrm{x}\right))}^2+(\mathrm{f}{\left(\mathrm{x}\right))}^3$

Differentiate the equation with respect to $\mathrm{x}$ we get,

$$\begin{gathered} \mathrm{h}\text{'}\left(\mathrm{x}\right)=\mathrm{f}\text{'}\left(\mathrm{x}\right)-2\mathrm{f}\left(\mathrm{x}\right).\mathrm{f}\text{'}\left(\mathrm{x}\right)+3(\mathrm{f}{\left(\mathrm{x}\right))}^2.\mathrm{f}\text{'}(\mathrm{x}) \\ =\mathrm{f}\text{'}(\mathrm{x})\left[1-2\mathrm{f}\left(\mathrm{x}\right)+3(\mathrm{f}{\left(\mathrm{x}\right))}^2\right] \\ =3\mathrm{f}\text{'}(\mathrm{x})\left[(\mathrm{f}{\left(\mathrm{x}\right))}^2-\frac{2}{3}\mathrm{f}(\mathrm{x})+\frac{1}{3}\right] \\ =3\mathrm{f}\text{'}(\mathrm{x})\left[(\mathrm{f}{\left(\mathrm{x}\right)-\frac{1}{3})}^2+\frac{1}{3}-\frac{1}{9}\right] \\ =3\mathrm{f}\text{'}(\mathrm{x})\left[{\left(\mathrm{f}\left(\mathrm{x}\right)-\frac{1}{3}\right)}^2+\frac{2}{9}\right] \end{gathered}$$

Here, if $\mathrm{f}\text{'}\left(\mathrm{x}\right)<0\Rightarrow \mathrm{h}\text{'}\left(\mathrm{x}\right)<0$ and if $\mathrm{f}\text{'}\left(\mathrm{x}\right)>0\Rightarrow \mathrm{h}\text{'}\left(\mathrm{x}\right)>0$

Therefore, $\mathrm{h}$ is increasing whenever $\mathrm{f}$ is increasing.

Hence, option A is the correct answer.