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Mathematics · Glossary

What is Local extremum?

Also known as: extremum

Definition 22.8 High School Mathematics · Chapter 22 — Differentiation and Convexity

ff has a local maximum at aa if f(x)≤f(a)f(x) \leq f(a) for all xx near aa; local minima are defined symmetrically.

At a local extremum inside an open interval, the tangent (orange) is horizontal: f'(a) = 0. Here f(x) = x3 - 3x, with a local maximum at -1 and a local minimum at 1.
At a local extremum inside an open interval, the tangent (orange) is horizontal: f′(a)=0f'(a) = 0. Here f(x)=x3−3xf(x) = x^3 - 3x, with a local maximum at −1-1 and a local minimum at 11.
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