Question

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`?

Answer 1

`f(x)` is increasing in the interval `(-oo,-2)`, reaches a maximum for `x=-2` then decreases in the interval `(-2,1)`, 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'(x) = 6x^2+6x-12`

First, we wind the points where `f'(x)=0` that are the critical points:

`6x^2+6x-12 = 0`

`x^2+x-2 = 0`

The roots are:

`x_1=-2`
`x_2 = 1`

Then we look at the sign of `f'(x)`: 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'(x) > 0` for `x in (-oo,-2)` and `x in (1,+oo)`

`f'(x) < 0` for `x in (-2,1)`

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

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