Find the intervals in which the following functions are increasing or decreasing.

Given:- Function $f(x)=x^{4}-4 x$

Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$.

(i) If $f^{\prime}(x)>0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$

(ii) If $f^{\prime}(x)<0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$

Algorithm:-

(i) Obtain the function and put it equal to $f(x)$

(ii) Find $f^{\prime}(x)$

(iii) Put $f^{\prime}(x)>0$ and solve this inequation.\

For the value of $x$ obtained in (ii) $f(x)$ is increasing and for remaining points in its domain it is decreasing.

Here we have,

$f(x)=x^{4}-4 x$

$\Rightarrow f^{\prime}(x)=\frac{d}{d x}\left(x^{4}-4 x\right)$

$\Rightarrow f^{\prime}(x)=4 x^{3}-4$

For $f(x)$ lets find critical point, we must have

$\Rightarrow f^{\prime}(x)=0$

$\Rightarrow 4 x^{3}-4=0$

$\Rightarrow 4\left(x^{3}-1\right)=0$

$\Rightarrow x=1$

clearly, $f^{\prime}(x)>0$ if $x>1$

and $f^{\prime}(x)<0$ if $x<1$

Thus, $f(x)$ increases on $(1, \infty)$

and $f(x)$ is decreasing on interval $x \in(-\infty, 1)$