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) = 3x⁴ - 4x³- 12x² + 5

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Answer

Consider the given function
f(x) = 3x⁴ - 4x³- 12x² + 5
⇒ f'(x) = 12x³ - 12x² - 24x
⇒ f'(x) = 12x(x² - x - 2)
⇒ f'(x) = 12x(x + 1)(x- 2)
For f(x) to be increasing, we must have,
f'(x) > 0
⇒ 12x(x + 1)(x- 2) > 0
⇒ x(x + 1)(x- 2) > 0
$\Rightarrow-1<\text{x}<0\text{ or }2<\text{x}<\infty$
$\Rightarrow\text{x}\in(-1,0)\cup(2,\infty)$
So, f(x) is increasing on $(-1,0)\cup(2,\infty).$
For f(x) to be decreasing, we must have,
f'(x) < 0
⇒ 12x(x + 1)(x- 2) < 0
⇒ x(x + 1)(x- 2) < 0
$\Rightarrow-\infty<\text{x}<-1\text{ or }0<\text{x}<2$
$\Rightarrow\text{x}\in(-\infty,-1)\cup(0,2)$
So, f(x) is decreasing in $(-\infty,-1)\cup(0,2).$

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