Question:
Find the value(s) of a for which $f(x)=x^{3}-a x$ is an increasing function on $R$ ?
Solution:
We have,
$f(x)=x^{3-a x}$
$f^{\prime}(x)=3 x^{2}-a$
Given that $f(x)$ is on increasing function
$\therefore \mathrm{f}^{\prime}(\mathrm{x}) 0$ for all $x \in \mathrm{R}$
$\Rightarrow 3 x^{2}-a>0$ for all $x \in R$
$\Rightarrow \mathrm{a}<3 \mathrm{x}^{2}$ for all $\mathrm{x} \in \mathrm{R}$
But the last value of $3 x^{2}=0$ for $x=0$
$\therefore a \leq 0$