Question

Let for a function $f\left(x\right)$, $h\left(x\right)=\left(f\right(x){)}^2+(f\left(x\right){)}^3$ 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)

The correct options are
A $h$ is increasing whenever $f$ is increasing
C $h$ is decreasing whenever $f$ is decreasing

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

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

Now

$h^{\prime }\left(x\right)>0$ implies

$f^{\prime }\left(x\right)\left(2f\right(x)+3f(x{)}^2)>0$

Or

$f^{\prime }\left(x\right)>0$ and $\left(2f\right(x)+3f(x{)}^2)>0$

Hence $h\left(x\right)$ is increasing if $f\left(x\right)$ is increasing and $h\left(x\right)$ is decreasing if $f\left(x\right)$ is decreasing.