Let for all real values of x, $h(x)=f(x)-\{f(x)\}^2+\{f(x)\}^3, x ∈ R$. Then

when f is increasing then h is decreasing

We know that f is increasing or decreasing according as or < 0. Also

$h'(x)=f'(x)-2f(x)+3\{f(x)\}^2f'(x)=f'(x)[3\{f(x)\}^2-2f(x)+1]$ .....(a)

Now for $3\{f(x)\}^2-2f(x)+1, B^2-4AC=4-12=-8<0$ and $A=3>0$, 

so $3\{f(x)\}^2-2f(x)+1>0∀x$

Hence $h'(x)$ is +ve or -ve according as $f'(x)>0$ or < 0

⇒ $h(x)$ is increasing when f is increasing  or

⇒ $h(x)$ is decreasing when f is decreasing