Maximum and Minimum of a Function

concepts in mathematical analysis. The value of a function at some point of a set on which this function is defined is said to be the maximum (minimum) on this set if the function does not have a greater (lesser) value at any other point of the set. If the value is a maximum (minimum) of the function for some neighborhood of the point, then we call it a local maximum (minimum).

The maximum and minimum of a function defined on an interval can occur at points at which the derivative vanishes, at points at which the derivative does not exist, or at the end points of the interval. A continuous function defined on a closed interval will inevitably have a maximum and minimum in that interval. However, if the continuous function is considered on an open interval (that is, an interval containing none of the end points), it may lack both a maximum and minimum on this interval. For example, the function y = x defined on the closed interval [0, 1] attains its maximum at x= 1 and its minimum at x = 0, that is, at the end points of the interval. However, if we consider this function on the open interval (0, 1), the function has neither a maximum nor a minimum on this interval, since for every x₀ a point can always be found in this interval that lies to the right (left) of x₀ and at which the value of the function will be greater (less) than at x₀ Similar assertions are true for functions of several variables.