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
• 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

$h^1\left(x\right)=f^1\left(x\right)-2f\left(x\right)f^1\left(x\right)+3f^2\left(x\right)f^1\left(x\right)$
$=f^1\left(x\right)\left(3f^2\right(x)+2f(x)+1)$
$=f^1\left(x\right)\left(2f^2\right(x)+(f\left(x\right)-1{)}^2)$
$h^1\left(x\right)>0$ If $f\left(x\right)$ is increasing
$\left(f^1\right(x)>0)$