Question

If $f^{\prime }\left(x\right)$ exists for all $x\in R$ and $g\left(x\right)=f\left(x\right)-\left(f\right(x){)}^2+(f\left(x\right){)}^3$ for all $x\in R$ , then

• A. $g\left(x\right)$ is increasing whenever $f$ is increasing
• B. $g\left(x\right)$ is increasing whenever $f$ is decreasing
• C. $g\left(x\right)$ is decreasing whenever $f$ is increasing
• D. none of these

Answer 1

The correct option is A $g\left(x\right)$ is increasing whenever $f$ is increasing

$g\left(x\right)=f\left(x\right)-\left(f\right(x){)}^2+(f\left(x\right){)}^3$

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

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

$=3f^{{}^{\prime }}\left(x\right)\left(\right(f\left(x\right)-\frac{1}{3}{)}^2+\frac{2}{9})$

The term in the bracket is always greater than zero.

So, $g^{{}^{\prime }}\left(x\right)=k{f}^{{}^{\prime }}\left(x\right)$ where $k>0$

So, $g\left(x\right)$ is increasing whenever $f\left(x\right)$ is increasing and $g\left(x\right)$ is decreasing whenever $f\left(x\right)$ is decreasing.

The answer is option (A).