Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=5x^3-15x^2-120x+3$

Answer 1

Given the function $f\left(x\right)=5x^3-15x^2-120x+3$.

Now $f^{\prime }\left(x\right)=15x^2-30x-120=15(x^2-2x-8)=15(x-4)(x+2)$.

Now

$f^{\prime }\left(x\right)\ge 0$ if $(x-4)(x+2)\ge 0$ then $x\ge 4,x\le -2$

So $f\left(x\right)$ is increasing on $(-\infty ,-2]\cup [4,\infty )$.

$f^{\prime }\left(x\right)<0$ if $(x-4)(x+2)\le 0\Rightarrow$ $-2\le x\le 4$

So $f\left(x\right)$ is decreasing on $[-2,4]$.