## Definition 6.5.11: Local Extremum

Let

*f*be a function defined on a domain*, and***D***c*a point in*.***D**- If there exists a neighborhood
of**U***c*with*f(c) f(x)*for all*x*in, then**U***f(c)*is called a**local maximum**for the function*f*that occurs at*x = c*. - If there exists a neighborhood
of**U***c*with*f(c) f(x)*for all*x*in, then**U***f(c)*is called a**local minimum**for the function*f*that occurs at*x = c*. - If
*f(x)*has either a local minimum or a local maximum at*x = c*, then*f(c)*is called**local extremum**of the function*f*.