Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=5+36x+3x^2-2x^3$

Answer 1

Given the function $f\left(x\right)=5+36x+3x^2-2x^3$.

Now,

$f^{\prime }\left(x\right)$

$=36+6x-6x^2$

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

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

So the function $f\left(x\right)$ is going to be an increasing function if $f^{\prime }\left(x\right)\ge 0$ i.e.

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

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

the function $f\left(x\right)$ is going to be an decreasing function 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 given function is increasing in $[-2,3]$ and decreasing in

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