Find the intervals in which the following functions are increasin… - Vidyadip

Question

Find the intervals in which the following functions are increasing or decreasing.$f(x) = x^4- 4x$

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Answer

$f(x) = x^4- 4x f'(x) = 4x^3- 4$
$= 4(x^3- 1)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 4(x^3- 1) > 0$
$\Rightarrow x^3- 1 > 0$
$\Rightarrow x^3> 1$
$\Rightarrow x^> 1$
$\Rightarrow\text{x}\in(1,\infty)$
So, f(x) is increasing on $(1,\infty).$
For f(x) to be decreasing, we must have
$f'(x) < 0$
$\Rightarrow 4(x^3- 1) < 0$
$\Rightarrow x^3- 1 < 0$
$\Rightarrow x^3< 1$
$\Rightarrow x^< 1$
$\Rightarrow\text{x}\in(-\infty,1)$
So, $f(x)$ is decreasing on $\text{x}\in(-\infty,1).$

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