# How do you find all intervals where the function f(x)=1/3x^3+3/2x^2+2 is increasing?

Sep 25, 2014

$f$ is increasing on $\left(- \infty , - 3\right]$ and $\left[0 , \infty\right)$.

The graph of $f$ looks like this:

Let us look at some details.

$f \left(x\right) = \frac{1}{3} {x}^{3} + \frac{3}{2} {x}^{2} + 2$

By solving $f ' \left(x\right) = 0$ for $x$,

$f ' \left(x\right) = {x}^{2} + 3 x = x \left(x + 3\right) = 0$,

we find the critical values: $x = - 3 , 0$.

Using the critical values to divide $\left(- \infty , \infty\right)$ into

$\left(- \infty , - 3\right]$, $\left[- 3 , 0\right]$, and $\left[0 , \infty\right)$

and we choose sample values $- 4$, $- 2$ and $1$ for the intervals above, respectively. (We can use any number on each interval excluding endpoints.)

$f ' \left(- 4\right) = 4 > 0 R i g h t a r r o w f$ is increasing on $\left(- \infty , - 3\right]$

$f ' \left(- 2\right) = - 2 < 0 R i g h t a r r o w f$ is decreasing on $\left[- 3 , 0\right]$

$f ' \left(1\right) = 4 > 0 R i g h t a r r o w f$ is increasing on $\left[0 , \infty\right)$