The first derivative test is a way to find if a critical point of a continuous function is a relative minimum or maximum. Simply, if the first derivative is negative to the left of the critical point, and positive to the right of it, it is a relative minimum. If the first derivative test finds the first derivative is positive to the left of the critical point, and negative to the right of it, the critical point is a relative maximum.
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.