Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=8+36x+3x^2-2x^3$

Answer 1

Given the function is

$f\left(x\right)=8+36x+3x^2-2x^3$.

Now,

$f^{\prime }\left(x\right)=36+6x-6x^2$

$=-6(x^2-x-6)$

$=(-6)(x-3)(x+2)$.

So the function $f\left(x\right)$ will be increasing if

$f^{\prime }\left(x\right)\ge 0$ i.e.

$(x-3)(x+2)\le 0$

or, $-2\le x\le 3$.

Also the function $f\left(x\right)$ will be decreasing if

$f^{\prime }\left(x\right)\le 0$ i.e.

$(x-3)(x+2)\ge 0$

or, $x\ge 3,x\le -2$.

So the function is increasing in $[-2,3]$ and decreasing on

$(-\infty ,-2]\cup [3,\infty )$.