Question

Find the interval in which the following function is increasing or decreasing.
$f\left(x\right)=\frac{x^4}{4}+\frac{2}{3}{x}^3-\frac{5}{2}{x}^2-6x+7$

Answer 1

$f\left(x\right)=\frac{x^4}{4}+\frac{2}{3}{x}^3-\frac{5}{2}{x}^2-6x+7$

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

$f^{\prime }\left(x\right)=\frac{4x^3}{4}+\frac{2}{3}\times 3x^2-\frac{5}{2}\times 2x-6$

$=x^3+2x^2-5x-6$

$=(x+1)(x^2+x-6)$ ................. (By hit and trial, we get (x+1) as its factor)

$=(x+1)(x+3)(x-2)$

Now make it sign scheme

$\therefore f\left(x\right)$ is decreasing in $(-\infty ,-3)\cup (-1,2)$

increasing in $(-3,-1)\cup (2,\infty )$