A physicist will often try to minimize energy, an economist cost, an engineer quality, and a merchant revenue. Such situations can generally be described by a real-valued function of several variables that must be optimized. This is a function f that, for each set of parameter values x1, *x 2, *x 3… *x n , returns a number denoted by f (x*1, *x 2, *x 3… *x n ), corresponding to the quantity to be optimized. The challenge is then to determine the parameter values for which f (x*1, *x 2, *x 3… *x n *) is minimal (or maximal, depending on the context).
Graph of a function of two variables.
A universal problem
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Put so broadly, the problem is far too vast to address in full. We might even say, with Euler, that it encompasses all of reality, "for since the fabric of the universe is most perfect and is the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear."
