
# 5.1: Maxima and Minima

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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.

Local maximum and minimum points are quite distinctive on the graph of a function, and are therefore useful in understanding the shape of the graph. In many applied problems we want to find the largest or smallest value that a function achieves (for example, we might want to find the minimum cost at which some task can be performed) and so identifying maximum and minimum points will be useful for applied problems as well. Some examples of local maximum and minimum points are shown in figure 5.1.1.

### Contributors

• Integrated by Justin Marshall.