Question

Let $h\left(x\right)=f\left(x\right)-\left(f\right(x){)}^2+(f\left(x\right){)}^3$ for every real number $x$, then

• A. $h$ is increasing whenever $f$ is increasing and decreasing whenever $f$ is decreasing
• B. $h$ is increasing whenever $f$ is decreasing
• C. $h$ is decreasing whenever $f$ is increasing
• D. Nothing can be said in general

Answer 1

The correct option is A $h$ is increasing whenever $f$ is increasing and decreasing whenever $f$ is decreasing

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

When $x=2$

$h\left(x\right)=2-2^2+2^3=6$

When $x=-2$

$h\left(x\right)=-2-(-2{)}^2+(-2{)}^3=-14$

Therefore, h is increasing whenever f is increasing and decreasing whenever f is decreasing.