(abbrev. max), n. 1. the largest element of a set S, usually denoted maxS; for example, the negative numbers have no maximum, but the non-positive numbers have maximum 0, although both have 0 as supremum. See maximal. 2. the highest value of a function, usually denoted max f; it is a global maximum if this condition is satisfied in comparison with all other values of the function. A local maximum is a value greater than any other in a neighborhood of its argument, and in the case of a real differentiable function on an open set is identified by having zero derivative and negative second derivative since the tangent to the curve changes from rising to falling at this point. In the figure below, the left-hand maximum is global and the other local. See first derivative test and second derivative test.
A is a global and B a local maximum.