Question

Let $h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$ for all real x

h(x) increases as f(x) increases for all x if

• A. $aϵ(0,3)$
  
• B. $ϵ[-2,2]$
  
• C. $aϵ[3,\infty )$
  
• D. $aϵ(5,\infty )$

Answer 1

The correct option is A $aϵ(0,3)$


$h\left(x\right)=f\left(x\right)-a\left(f\right(x){)}^2+a(f\left(x\right){)}^3$
$h^{\prime }\left(x\right)=f^{\prime }\left(x\right)\left[3a\right(f\left(x\right){)}^2-2a\left(f\right(x\left)\right)+1]$
Now $h^{\prime }\left(x\right)$ >$0$ if $f^{\prime }\left(x\right)$ > $0$ and $3a\left(f\right(x){)}^2-2af(x)+1$ > $0$
$\Rightarrow 3a$ >$0,\Delta$ < $0\Rightarrow 4a^2-12a$ < $0$
$a$ > $0$ $aϵ(0,3)$
$h^{\prime }\left(x\right)$ > $0$ if $f^{\prime }\left(x\right)$ < $0$ and $3a\left(f\right(x){)}^2-2af(x)+1$ < $0$
$\Rightarrow 3a$ < $0$ and $\Delta$ < $0$
$\Rightarrow a$ < $0$ and $aϵ(0,3)$.
No value of a