Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=x^3-6x^2-36x+2$

Answer 1

Given the function is $f\left(x\right)=x^3-6x^2-36x+2$.

Now,

$f^{\prime }\left(x\right)=3x^2-12x-36=3(x^2-4x-12)=3(x-6)(x+2)$.

Now, $f^{\prime }\left(x\right)\ge 0$ if

$(x-6)(x+2)\ge 0$

or, $x\ge 6$, $x\le -2$

This gives $f\left(x\right)$ will be increasing in

$(-\infty ,-2]\cup [6,\infty )$.

Again the function will be decreasing if,

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

or, $(x-6)(x+2)\le 0$

or, $x\in [-2,6]$.

So the function is decreasing in $[-2,6]$.