Increasing and Decreasing Functions

In the previous section, we discussed four important theorems in elementary differential calculus. In this section we are going to define increasing and decreasing functions and how these terms relate to the derivative.

You can call a function an increasing function on an interval under the following condition: If for all x in this interval where
< , then < .

You can call a function a decreasing function on an interval under the following condition: If for all x in this interval where
< , then > .

You can find increasing functions also using the derivative. If f'(x) > 0 for all x in this interval, then the function is an increasing function. The same works for decreasing functions. If f'(x) < 0 for all x in an interval then the function is a decreasing function. Why does this work? Remember that the derivative tells us the slope. A positive slope means that as the x-coordinate increases the y-coordinate also increases causing an "uphill". A negative slope means that as we go to the right of the graph of the function, the y-variable(or f(x)) is decreasing causing a "downhill".

Let's practice classifying functions as increasing or decreasing. Suppose we had a function that looked like this: . We can see the graph of f(x) below:


You can see on the interval [-3,3] as the x-coordinate increases the y coordinate or f(x) is also increasing. We can check our work by taking the derivative at any point on the interval. Suppose we check at x = 2. The derivative is . At x = 2, f'(x)=+12. The slope is positve so that tells us that it is an increasing function. is also an increasing function at x = -2 because the square of a negative number is a positive number.