# How do you determine where the function is increasing or decreasing, and determine where relative maxima and minima occur for 2x^3 + 3x^2 - 12x?

Dec 5, 2016

$f \left(x\right)$ is increasing in the interval $\left(- \infty , - 2\right)$, reaches a maximum for $x = - 2$ then decreases in the interval $\left(- 2 , 1\right)$, reaches a minimum in $x = 1$, then increases indefinitely.

#### Explanation:

You determine intervals of increasing and decreasing and the relative maxima and minima by studying the first derivative of the function:

$f ' \left(x\right) = 6 {x}^{2} + 6 x - 12$

First, we wind the points where $f ' \left(x\right) = 0$ that are the critical points:

$6 {x}^{2} + 6 x - 12 = 0$

${x}^{2} + x - 2 = 0$

The roots are:

${x}_{1} = - 2$
${x}_{2} = 1$

Then we look at the sign of $f ' \left(x\right)$: as this is a second order polynomial with positive leading coefficient, we know that it is negative inside the interval between the two roots, and positive outside, that is:

$f ' \left(x\right) > 0$ for $x \in \left(- \infty , - 2\right)$ and $x \in \left(1 , + \infty\right)$

$f ' \left(x\right) < 0$ for $x \in \left(- 2 , 1\right)$

So we can state that $f \left(x\right)$ is increasing in the interval $\left(- \infty , - 2\right)$, reaches a maximum for $x = - 2$ then decreases in the interval $\left(- 2 , 1\right)$, reaches a minimum in $x = 1$, then increases indefinitely.

graph{2x^3+3x^2-12x [-8, 8, -40, 40]}