This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# Maximize

 Maximizemaximizes f with respect to x. Maximizemaximizes f with respect to x, y, .... Maximizemaximizes f subject to the constraints cons. Maximizemaximizes with variables over the domain dom, typically Reals or Integers.
• cons can contain equations, inequalities, or logical combinations of these.
• If f and cons are linear or polynomial, Maximize will always find a global maximum.
• 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.
• Integers can be used to specify that a particular variable can take on only integer values.
Maximize a univariate function:
Maximize a multivariate function:
Maximize a function subject to constraints:
A maximization problem containing parameters:
Maximize a univariate function:
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Maximize a multivariate function:
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Maximize a function subject to constraints:
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A maximization problem containing parameters:
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 Scope   (15)
Unconstrained univariate polynomial maximization:
Constrained univariate polynomial maximization:
Univariate transcendental maximization:
Univariate piecewise maximization:
Multivariate linear constrained maximization:
Linear-fractional constrained maximization:
Unconstrained polynomial maximization:
Constrained polynomial optimization can always be solved:
The maximum value may not be attained:
The objective function may be unbounded:
There may be no points satisfying the constraints:
Algebraic maximization:
Bounded transcendental maximization:
Piecewise maximization:
Unconstrained parametric maximization:
Constrained parametric maximization:
Integer linear programming:
Polynomial maximization over the integers:
 Options   (1)
Finding the exact solution takes a long time due to high-degree algebraic number operations:
With WorkingPrecision, you get an exact maximum value, but it might be incorrect:
 Applications   (3)
Find the maximal area among rectangles with a unit perimeter:
Find the maximal area among triangles with a unit perimeter:
Find the maximum height reached by a projectile:
Find the maximum range of a projectile:
Maximize gives an exact global maximum of the objective function:
NMaximize attempts to find a global maximum numerically, but may find a local maximum:
FindMaximum finds local maxima depending on the starting point:
The maximum point satisfies the constraints, unless messages say otherwise:
The given point maximizes the distance from the point .
When the maximum is not attained, Maximize may give a point on the boundary:
Here the objective function tends to the maximum value when y tends to infinity:
Maximize can solve linear programming problems:
LinearProgramming can be used to solve the same problem given in matrix notation:
This computes the maximum value:
Maximize requires that all functions present in the input be real-valued:
Values for which the equation is satisfied but the square roots are not real are disallowed: