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Critical points exist where the derivative is zero or when the derivative does not exist (i.e. at discontinuities/cusps). Relative extrema are relative maximums or minimums in a function. These extrema occur at critical points, but not all critical points are relative extrema. When the derivative is equal to 0, we have three possible outcomes. The point is either a minimum, a maximum or a saddlepoint (see below).
The final word on relative extrema: for a point to be a relative extrema, the function must be smaller on both sides of the point or greater on both sides. When in doubt, test values barely to the right and barely to the left to test the points.
For more examples see here:
Khan Academy Quiz #1
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