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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 6+ 12x + 3x^2 - 2x^3$

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Answer

$f(x) = 6+ 12x + 3x^2 - 2x^3$
$f'(x) = 12 + 6x - 6x^2$
$= -6(x^2 - x - 2)$
$= -6(x - 2)(x + 1)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow -6(x - 2)(x + 1) > 0$
$\Rightarrow (x - 2)(x + 1) < 0$
$[$Since,$ -6 < 0, -6(x - 2)(x + 1) > 0 \Rightarrow (x - 2)(x + 1) < 0]$
$\Rightarrow -1 < x < 2$
$\Rightarrow\text{x}\in(-1,2)$
So, $f(x)$ is increasing on $(-1, 2).$
For $f(x)$ to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow -6(x - 2)(x + 1) < 0$
$\Rightarrow (x - 2)(x + 1) < 0$
$[$Since, $ -6 < 0, -6(x - 2)(x + 1) > 0 \Rightarrow (x - 2)(x + 1) > 0]$
$\Rightarrow x < -1$ or $x > 2$
$\Rightarrow\text{x}\in(-\infty,-1)\cup(2,\infty)$
So, $f(x)$ is decreasing on $(-\infty,-1)\cup(2,\infty).$

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