Question

Determine the intervals in which $f$ is inreasing and decreasing
$f:R\to R,f\left(x\right)=3x^4-4x^3-12x^2+5$

Answer 1

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

$f^{\prime }\left(x\right)=12x^3-12x^2-24x$

$f^{\prime }\left(x\right)=12(x^3-x^2-2x)$

$f^{\prime }\left(x\right)=12x(x+1)(x-2)$

The critical point are $x=-1,0,2$

1) In $(-\infty ,-1),f^{\prime }\left(x\right)<0$ and hence $f\left(x\right)$ is decreasing.

2) In $(-1,0),f^{\prime }\left(x\right)>0$ and hence $f\left(x\right)$ is increasing.

3) In $(0,2),f^{\prime }\left(x\right)<0$ and hence $f\left(x\right)$ is decreasing.

4) In $(2,\infty ),f^{\prime }\left(x\right)>0$ and hence $f\left(x\right)$ is increasing.