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Easy: Critical points
Critical points are where f′(x) = 0 (or doesn't exist). Between them f′ keeps one sign: where f′ > 0 the function increases, where f′ < 0 it decreases. A critical point where f′ changes from + to − is a local maximum; from − to + it is a local minimum.
Use the first derivative to answer each question.
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No. f′ must change sign there. f(x) = x³ has a critical point at 0, but f′ is positive on both sides, so it's neither.
At a critical point, f″ > 0 means the graph is concave up, so it's a local minimum; f″ < 0 means a local maximum. If f″ = 0 the test is inconclusive.
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