Question

Suppose $f^{\prime }\left(x\right)$ exists for each $x$ and $h\left(x\right)=f\left(x\right)-\left(f\right(x){)}^2+(f\left(x\right){)}^3$ for every real number $x$. Then

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

Answer 1

Answer: (A), (C)

$h$ is increasing whenever f is increasing
$h$ is decreasing whenever f is decreasing

$h\left(x\right)=f\left(x\right)-\left(f\right(x){)}^2+(f\left(x\right){)}^3$
$h^{\prime }\left(x\right)=f^{\prime }\left(x\right)-2f^{\prime }\left(x\right)f\left(x\right)+3f^{\prime }\left(x\right)f(x{)}^2$
$=3f^{\prime }\left(x\right)\left[f\right(x{)}^2-\frac{2}{3}f\left(x\right)+\frac{1}{3}]$
$=3f^{\prime }\left(x\right)[{\left(f\right(x)-1/3)}^2+2/9]$
Thus, $h^{\prime }\left(x\right)>0$ if $f^{\prime }\left(x\right)>0$
and $h^{\prime }\left(x\right)<0$ if $f^{\prime }\left(x\right)<0$
Therefore, $h$ increases whenever $f$ increases
and $h$ decreases whenever $f$ decreases
Ans: A,C