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A local maximum point on a function is a point $$(x,y)$$ on the graph of the function whose $$y$$ coordinate is larger than all other $$y$$ coordinates on the graph at points "close to'' $$(x,y)$$. More precisely, $$(x,f(x))$$ is a local maximum if there is an interval $$(a,b)$$ with $$a < x < b$$ and $$f(x)\ge f(z)$$ for every $$z$$ in $$(a,b)$$. Similarly, $$(x,y)$$ is a local minimum point if it has locally the smallest $$y$$ coordinate. Again being more precise: $$(x,f(x))$$ is a local minimum if there is an interval $$(a,b)$$ with $$a < x < b$$ and $$f(x)\le f(z)$$ for every $$z$$ in $$(a,b)$$. A local extremum is either a local minimum or a local maximum.