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

Question

Find the intervals in which the following functions are increasing or decreasing. $f(x) = (x - 1)(x - 2)^2$

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Answer

$f(x) = (x - 1)(x - 2)^2$
$= (x - 1)(x^2 - 4x + 4)$
$= x^3 - 5x^2+ 8x - 4$
$f'(x) = 3x^2 - 10x + 8$
$= 3x^2 - 6x - 4x + 8$
$= (x - 2)(3x - 4)$
For $f(x)$ to be increasing, we must have
$f'(x) > 0$
$\Rightarrow (x - 2)(3x - 4) > 0$
$\Rightarrow\text{x}<\frac{4}{3}\text{ or }\text{x}>2$
$\Rightarrow\text{x}\in\Big(-\infty,-\frac{4}{3}\Big)\cup(2,\infty)$
So, $f(x)$ is increasing on $\text{x}\in\Big(-\infty,-\frac{4}{3}\Big)\cup(2,\infty).$
For $f(x)$ to be decreasing, we must have,
$f'(x) < 0$
$\Rightarrow (x - 2)(3x - 4) < 0$
$\Rightarrow\frac{4}{3}<\text{x}<2$
$\Rightarrow\text{x}\in\Big(\frac{4}{3},2\Big)$
So, $f(x)$ is decreasing on $\text{x}\in\Big(\frac{4}{3},2\Big).$

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