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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^3 - 6x^2 - 36x + 2$

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Answer

$f(x) = x^3 - 6x^2 - 36x + 2$
$\therefore$ $f'(x) = 3x^2 - 12x - 36$
Critical point
$f'(x) = 0$
$\Rightarrow 3(x^2 - 4x - 12) = 0$
$\Rightarrow (x - 6)(x + 2) = 0$
$\Rightarrow x = 6, -2$
Clearly, $f'(x) > 0$ if $x < -2$ and $x > 6$
$f'(x) < 0$ if $-2x < x < 6$
Thus, $f(x)$ increases in $(-\infty,-2)\cup(6,\infty),$ decreases in $(-2, 6)$.

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