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) = x³- 12x² + 36x + 17

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Answer

f(x) = x³- 12x² + 36x + 17
f'(x) = 3x² - 24x + 36
= 3(x² - 8x + 12)
= 3(x - 2)(x - 6)
For f(x) to be increasing, we must have
f'(x) > 0
⇒ 3(x - 2)(x - 6) > 0
⇒ (x - 2)(x - 6) > 0
[Since, 3 > 0, 3(x - 2)(x - 6) > 0 ⇒ (x - 2)(x - 6) > 0]
⇒ x < 2 or x > 6
$\Rightarrow\text{x}\in(-\infty,2)\cup(6,\infty)$
So, f(x) is increasing on $\text{x}\in(-\infty,2)\cup(6,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 3(x - 2)(x - 6) < 0
⇒ (x - 2)(x - 6) < 0
[Since, 3 > 0, 3(x - 2)(x - 6) < 0 ⇒ (x - 2)(x - 6) < 0]
⇒ 2 < x < 2
$\Rightarrow\text{x}\in(2,6)$
So, f(x) is decreasing on $\text{x}\in(2,6).$

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