Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSIncreasing and Decreasing Functions5 Marks
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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