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Geometrically, at a critical point where f'(p) = o, the line tangent to the graph is f at p is horizontal. At a critical point where f'(p) is undefined, there is not horizontal tangent to the graph - there is either a vertical tangent or not tangent at all. (For example, x = 0 is a critical point for the absolute value function f(x) = |x|.) However, most of the functions we will work with will be differentiable everywhere, and therefor most of our critical points will be of the f'(p) = 0 variety. The critical points divide the domain of f into intervals on which the sign of the derivative remains the same, either positive or negative. Therefor, if f is defined on the interval between two successive critical points, it's graph can not change direction on that interval; it is either increasing or decreasing.Theorem: If a continuous function f has a local maximum or minimum at p, and if p is not an endpoint of the domain, then p is a critical point.
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