Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=\frac{x^4}{4}-x^3-5x^2+24x+12$

Answer 1

$f\left(x\right)=\frac{x^2}{4}-x^3-5x^2+24x+12$

$f^{\prime }\left(x\right)=\frac{4x^3}{4}-3x^2-10x+24$

$=x^3-3x^2-10x+24$

$=(x-2)(x^2-x-12)$ (by hit and trial)

$=(x-2)(x-4)(x+3)$

for incresing/decreasing $f^{\prime }\left(x\right)\ge 0$

Drawing its sign scheme we have

$\therefore f\left(x\right)$ is increasing in $(-3,2)\cup (4,\infty )$

$f\left(x\right)$ is decreasing in $(-\infty ,-3)\cup (2,4)$