Find the value(s) of a for which
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 $x=0$

$\therefore a \leq 0$

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