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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 2x^3 + 9x^2 + 12x + 20$

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Answer

$f(x) = 2x^3 + 9x^2 + 12x + 20$
$f'(x) = 6x^2 + 18x + 12$
$= 6(x^2 + 3x + 2)$
$= 6(x + 1)(x + 2)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 6(x + 1)(x + 2) > 0$
$\Rightarrow (x + 1)(x + 2) > 0$
$[$Since, $6 > 0, 6(x + 1)(x + 2) > 0 \Rightarrow (x + 1)(x + 2) > 0]$
$\Rightarrow x < -2$ or $x > -1$
$\Rightarrow\text{x}\in(-\infty,-2)\cup(-1,\infty)$
So, $f(x)$ is increasing on $\text{x}\in(-\infty,-2)\cup(-1,\infty).$
For $f(x)$ to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow 6(x + 1)(x + 2) < 0$
$\Rightarrow (x + 1)(x + 2) < 0$
$[$Since, $6 > 0, 6(x + 1)(x + 2) < 0 \Rightarrow (x + 1)(x + 2) < 0]$
$\Rightarrow -2 < x < -1$
$\Rightarrow\text{x}\in(-2,-1)$
So, $f(x)$ is decreasing on $\text{x}\in(-2,-1).$

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