Finding intervals of increase and decrease of a function can be done using either a graph of the function or its derivative. These intervals of increase and decrease are important in finding critical points, and are also a key part of defining relative maxima and minima and inflection points.
I want to talk about how to determine when a function is increasing or decreasing let's take a look at a graph of just sort of generic function y equals f of x and I've drawn some tangent lines in a few places. Just looking at these tangent lines you'll notice that some of these tangents lines have positive slopes some have negative slopes and that's going to be the key to determining whether a function is increasing or decreasing over an interval. So you might observe that if f prime of x is positive then the tangent line slopes up, right because f prime gives us the slope of the tangent line at a given point.
Now here f prime would be negative and so the tangent line will slope down, and this tells us where the functions increases or decreases where the tangent line slope up the functions increasing and where the tangent line slope down the function is decreasing, so you could see that this function increases in 2 intervals for x less than a for x greater than b and it decreases between a and b. So let's write a theorem we'll call this the increasing, decreasing test and so in the future when I refer to the increasing, decreasing test I mean this theorem. If f prime is positive on an interval then f increases on that interval, and if f prime is negative on an interval then f decreases.
This is a very important theorem and we'll refer to it a lot when we do our curve sketching in the future but the next few exercises will focus on using this theorem to determine where our function increases or decreases.