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

Question

Find the intervals in which the following functions are increasing or decreasing.
$f(x) = 3x^4 - 4x^3- 12x^2 + 5$

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Answer

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

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