The First Derivative Test for Relative Maximum and Minimum - Concept

So we just learned about relative maxima and minima and I want to show you a test that shows how you can find them. The first derivative test and it starts out assuming that f is a continuous function and c is one of its critical points. If f prime is negative to the left of c and positive to the right then f has a relative minimum at c and if that prime is positive to the left of c and negative to the right then f has a relative maximum at c. Now let's see why this is true, I have drawings of this situation here, the first situation remember f prime was negative to the left of c and positive to the right. That means that the function f is decreasing to the left and increasing to the right so it makes sense that you should have a relative minimum here.
But remember what we're going to be searching for is where the derivative goes from negative to positive that will indicate a relative minimum. And here is the opposite situation you have the derivative going from positive to the left of c to negative to the right and the middle you've got a relative maximum. So we'll be looking for the function that go from positive and negative that indicates that it's going from increase into decreasing so that's why we get a relative maximum. This is called the first derivative test for relative maxima and minima.