Increasing and Decreasing f(x)

Since the derivative of a function gives the equation of slope of the function, you can use the derivative to determine on which intervals f(x) is increasing or decreasing.

Here's an example:
Given f(x) = 1/3x^3 + 5/2x^2+4x+9, determine on which intervals f(x) is increasing or decreasing.

First find the derivative.
f'(x) = x^2+5x+4
Now, find the zeros, or rather the critical points where the sign may change. We use this cause the function can't cross the x-axis without changing sign.
f'(x) = (x+4)(x+1), thus the zeros are -4 and -1.
Use a sign graph.

The sign for the intervals (-inf,-4) and (-1,+inf) are +, so the function is increasing on these intervals.
The sign for the interval (-4,-1) is -, so the function is decreasing on this interval.