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Maximize

Global symbolic maximization of a function over variables with optional constraints.

$ wo 'Maximize[x^2 - 4*x + 5, x]'

Maximize::natt: The maximum is not attained at any point satisfying the given constraints\.\s? (regex)
{Infinity, {x -> -Infinity}}

Constraints that leave the objective unbounded report the same way.

$ wo 'Maximize[{x^2, x > 1}, x]'

Maximize::natt: The maximum is not attained at any point satisfying the given constraints\.\s? (regex)
{Infinity, {x -> Infinity}}

A bounded region gives an exact optimum, including for a periodic objective whose critical points form a family.

$ wo 'Maximize[{Sin[x], 0 < x < 2*Pi}, x]'
{1, {x -> Pi/2}}

A third argument names the domain. Reals is the default, so it changes nothing:

$ wo 'Maximize[-x^2 + 4 x, x, Reals]'
{4, {x -> 2}}

Over Integers, a real optimum that is already an integer carries over — it is feasible there, and nothing integral can beat it:

$ wo 'Minimize[x^2 - 4 x, x, Integers]'
{-4, {x -> 2}}

$ wo 'Maximize[x^2, x, Integers]'

Maximize::natt: The maximum is not attained at any point satisfying the given constraints\.\s? (regex)
{Infinity, {x -> -Infinity}}