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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = x^3 - 6x^2 + 9x + 15$

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Answer

$f(x) = x^3 - 6x^2 + 9x + 15$
$f'(x) = 3x^2 - 12x + 9$
$= 3(x^2 - 4x + 3)$
$= 3(x - 1)(x - 3)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow 3(x - 1)(x - 3) > 0$
$\Rightarrow (x - 1)(x - 3) > 0$
$[$Since, $3 > 0, 3(x - 1)(x - 3) > 0 \Rightarrow (x - 1)(x - 3) > 0]$
$\Rightarrow x < 1$ or $x > 3$
$\Rightarrow\text{x}\in(-\infty,1)\cup(3,\infty)$
So, $f(x)$ is increasing on $\text{x}\in(-\infty,1)\cup(3,\infty).$
For $f(x)$ to be decreasing, we must have,
f'(x) < 0
$\Rightarrow 3(x - 1)(x - 3) < 0$
$\Rightarrow (x - 1)(x - 3) < 0$
$[$Since, $3 > 0, 3(x - 1)(x - 3) < 0 \Rightarrow (x - 1)(x - 3) < 0]$
$\Rightarrow 1 < x > 3$
$\Rightarrow\text{x}\in(1,3)$
So, $f(x)$ is decreasing on $\text{x}\in(1,3).$

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