Built-in Mathematica Symbol

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[{f, cons}, {x, y, ...}, dom]
maximizes with variables over the domain dom, typically Reals or Integers.

MORE INFORMATION

Maximize returns a list of the form {f_max, {x->x_max, y->y_max, ...}}.

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.

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, ...}}.

EXAMPLES

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Basic Examples (4)

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->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 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:

Possible Issues (1)

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: