Question

Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
for every real number $x$.
If $f\left(x\right)$ is strictly increasing function, then $h\left(x\right)$ is non-monotonic function given

• A. $a\in (0,3)$
• B. $a\in (-2,2)$
• C. $a\in (3,\infty )$
• D. $a\in (-\infty ,0)\cup (3,\infty )$

Answer 1

The correct option is D $a\in (-\infty ,0)\cup (3,\infty )$

$h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
or $h^{\prime }\left(x\right)=f^{\prime }\left(x\right)-2af\left(x\right)f^{\prime }\left(x\right)+3a\left(f\right(x\left){)}^2{f}^{\prime }\right(x)$
$=f^{\prime }\left(x\right)\left[3a\right(f\left(x\right){)}^2-2af\left(x\right)+1]$

Now, $h\left(x\right)$ is non monotonic function if
$3a\left(f\right(x){)}^2-2af(x)+1$ changes sign
So,
$D>0$
$\Rightarrow 4a^2-12a>0$
$\Rightarrow a\in (-\infty ,0)\cup (3,\infty )$