In mathematics, maxima and minima, known collectively as extrema, are the largest value (maximum) or smallest value (minimum), that a function takes in a point either within a given neighbourhood (local extremum) or on the function domain in its entirety (global extremum). Definitions A real-valued function f' defined on the real line is said to have a 'local maximum point at the point x^∗, if there exists some ε > 0, such that f(x^∗) ≥ f(x) when |x ? x^∗| < ε. The value of the function at this point is called maximum of the function. On a graph of a function, its local maxima will look like the tops of hills. Similarly, a function has a local minimum point at x^∗