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The First Derivative Test for Relative Maximum and Minimum - Concept
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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.

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