# MaxValue

MaxValue[f,x]

gives the maximum value of f with respect to x.

MaxValue[f,{x,y,}]

gives the maximum value of f with respect to x, y, .

MaxValue[{f,cons},{x,y,}]

gives the maximum value of f subject to the constraints cons.

MaxValue[,xreg]

constrains x to be in the region reg.

MaxValue[,,dom]

constrains variables to the domain dom, typically Reals or Integers.

# Details and Options • MaxValue[] is effectively equivalent to First[Maximize[]].
• MaxValue gives the supremum of values of f. It may not be attained for any values of x, y, .
• cons can contain equations, inequalities or logical combinations of these.
• The constraints cons can be any logical combination of:
•  lhs==rhs equations lhs!=rhs inequations lhs>rhs or lhs>=rhs inequalities {x,y,…}∈reg region specification Exists[x,cond,expr] existential quantifiers
• If f and cons are linear or polynomial, MaxValue will always find a global maximum.
• MaxValue[{f,cons},xreg] is effectively equivalent to MaxValue[{f,consxreg},x].
• For xreg, the different coordinates can be referred to using Indexed[x,i].
• MaxValue will return exact results if given exact input.
• If MaxValue is given an expression containing approximate numbers, it automatically calls NMaxValue.
• If no domain is specified, all variables are assumed to be real.
• xIntegers can be used to specify that a particular variable can take on only integer values.
• If the constraints cannot be satisfied, MaxValue returns .
• N[MaxValue[]] calls NMaxValue for optimization problems that cannot be solved symbolically.

# Examples

open allclose all

## Basic Examples(5)

Find the maximum value of a univariate function:

 In:= Out= Find the maximum value of a multivariate function:

 In:= Out= Find the maximum value of a function subject to constraints:

 In:= Out= Find the maximum value as a function of parameters:

 In:= Out= Find the maximum value of a function over a geometric region:

 In:= Out= 