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)(x² - 4x + 4)
= x³ - 5x²+ 8x - 4
f'(x) = 3x² - 10x + 8
= 3x² - 6x - 4x + 8
= (x - 2)(3x - 4)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ (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
⇒ (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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