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

Question

Find the intervals in which the following functions are increasing or decreasing. $f(x) = 5 + 36x + 3x^2- 2x^3$

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Answer

$f(x) = 5 + 36x + 3x^2- 2x^3$
$\therefore f'(x) = 36 + 6x - 6x^2$
Critical point
$f'(x) = 0$
$\Rightarrow 36 + 6x - 6x^2 = 0$
$\Rightarrow -6(x^2 - x - 6) = 0$
$\Rightarrow (x - 3)(x + 2) = 0$
$\therefore x = 3, -2$
Clearly $f'(x) > 0 $ if $-2 < x < 3$
Also $f'(x) < 0$ if $x < -2$ and $x > 3$
Thus increases if $\text{x}\in(-2,3),$ decreases if $\text{x}\in(-\infty,-2)\cup(3,\infty)$

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