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The critical points of a function are the points at which the function changes direction from increasing to decreasing, or vice versa. Critical points are valuable because they will be the points at which the function has a local or global maximum, or a local or global minimum. In other words, the extrema of a function will always be located at critical points, so if we can find the critical points of a function, we'll also be able to find the function's extrema.
Luckily, to find critical points, all we have to do is take the first derivative of the function, set it equal to 0, and solve for x. This will give us any potential critical points. The tricky part is that the points we find with this method are only potential critical points. In order to prove that they are in fact critical points, we have to prove that the function changes direction at those points, which we'll do using the first derivative test.
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