Maximize

Maximize[f,x]
maximizes f with respect to x.

Maximize[f,{x,y,}]
maximizes f with respect to x, y, .

Maximize[{f,cons},{x,y,}]
maximizes f subject to the constraints cons.

Maximize[,xreg]
constrains x to be in the region reg.

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

Details and OptionsDetails and Options

  • Maximize returns a list of the form .
  • cons can contain equations, inequalities, or logical combinations of these.
  • The constraints cons can be any logical combination of:
  • lhs==rhsequations
    lhs!=rhsinequations
    or inequalities
    {x,y,}regregion specification
    Exists[x,cond,expr]existential quantifiers
  • If f and cons are linear or polynomial, Maximize will always find a global maximum.
  • Maximize[{f,cons},xreg] is effectively equivalent to Maximize[{f,consxreg},x].
  • For , the different coordinates can be referred to using Indexed[x,i].
  • Maximize will return exact results if given exact input.
  • If Maximize is given an expression containing approximate numbers, it automatically calls NMaximize.
  • If the maximum is achieved only infinitesimally outside the region defined by the constraints, or only asymptotically, Maximize will return the supremum and the closest specifiable point.
  • 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, Maximize returns {-Infinity,{x->Indeterminate,}}.
  • N[Maximize[]] calls NMaximize for optimization problems that cannot be solved symbolically.
  • Maximize[f,x,WorkingPrecision->n] uses n digits of precision while computing a result. »

ExamplesExamplesopen allclose all

Basic Examples  (5)Basic Examples  (5)

Maximize a univariate function:

In[1]:=
Click for copyable input
Out[1]=

Maximize a multivariate function:

In[1]:=
Click for copyable input
Out[1]=

Maximize a function subject to constraints:

In[1]:=
Click for copyable input
Out[1]=

A maximization problem containing parameters:

In[1]:=
Click for copyable input
Out[1]=

Maximize a function over a geometric region:

In[1]:=
Click for copyable input
Out[1]=

Plot it:

In[2]:=
Click for copyable input
Out[2]=
Introduced in 2003
(5.0)
| Updated in 2014
(10.0)