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 $x$.
$h\left(x\right)$ increases as $f\left(x\right)$ decreases for all real values of $x$ if

• A. $a\in (0,3)$
• B. $a\in (-2,2)$
• C. $a\in (3,\infty )$
• D. None of these

Answer 1

The correct option is D None of these

$h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
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, $h\left(x\right)$ increases as $f\left(x\right)$ decreases for all real values of $x$ if
$3a\left(f\right(x){)}^2-2af(x)+1<0$ for all $x\in R$
So,
$3a<0$ and $4a^2-12a\le 0$
$a<0$ and $a\in [0,3]$
So, no such $a$ is possible