Built-in Mathematica Symbol

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[{f, cons}, {x, y, ...}, dom]
gives the maximum value of f over the domain dom, typically Reals or Integers.

MORE INFORMATION

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.

If f and cons are linear or polynomial, MaxValue will always find a global maximum.

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 -Infinity.

EXAMPLES

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Basic Examples (4)

Find the maximum value of a univariate function:

Find the maximum value of a multivariate function:

Find the maximum value of a function subject to constraints:

Find the maximum value as a function of parameters:

Find the maximum value of a univariate function:

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Find the maximum value of a multivariate function:

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Find the maximum value of a function subject to constraints:

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Find the maximum value as a function of 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 maximum can take a long time:

With WorkingPrecision->200, 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:

Properties & Relations (3)

Maximize gives both the value of the maximum and the maximizer point:

For strict polynomial inequality constraints MaxValue may be much faster than Maximize:

MaxValue gives an exact global maximum value of the objective function:

NMaxValue attempts to find a global maximum numerically, but may find a local maximum:

FindMaxValue finds local maxima depending on the starting point:

MaxValue can solve linear programming problems:

LinearProgramming can be used to solve the same problem given in matrix notation:

Possible Issues (1)

MaxValue 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: