Maximize

The global maximum of f over the real line (or a simple interval constraint), as {value, {x -> point}} -- an exact, provably global answer, not a local one.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageMaximize!↗

Maximize(f, x)the global maximum of f, a real expression in the single variable x, over f's own natural domain.

Maximize(f, {x})same as Maximize(f, x) -- Wolfram accepts the variable either bare or wrapped in a one-element list.

Maximize({f, cons}, x)the global maximum of f restricted to cons, a simple interval constraint (x >= a, a <= x <= b, an And of two one-sided bounds, ...).

Typed signatures

Type	Package	Applies
(expression, list<symbol> | symbol) -> expression	analytic	always

Domain: Calculus

Details

• Minimize's exact mirror image -- same scope, same machinery (D/Factor/Solve/Limit on the ORIGINAL expression), same declines. See its details for the full account; this entry only lists Maximize's own examples.

Examples

Basic #

A downward parabola's vertex is its global maximum

live ↗

In$\mathrm{Maximize}(-x^2+4x-1, x)$Out$\bigl\lbrack3, \bigl\lbrack\mathrm{Rule}(x, 2)\bigr\rbrack\bigr\rbrack$

An odd-degree polynomial is unbounded above

live ↗

In$\mathrm{Maximize}(x^3, x)$Out$\bigl\lbrack∞, \bigl\lbrack\mathrm{Rule}(x, ∞)\bigr\rbrack\bigr\rbrack$

With 0 <= x <= 1, the unconstrained vertex (x = 2) is out of range -- the boundary wins

live ↗

In$\mathrm{Maximize}(\bigl\lbrack-x^2+4x-1, x≥0∧x≤1\bigr\rbrack, x)$Out$\bigl\lbrack2, \bigl\lbrack\mathrm{Rule}(x, 1)\bigr\rbrack\bigr\rbrack$

Returns the maximum $\frac{49}{8}$ and where it occurs, $x=-\frac34$

live ↗

In$\mathrm{Maximize}(-2x^2-3x+5, x)$Out$\bigl\lbrack\frac{49}{8}, \bigl\lbrack\mathrm{Rule}(x, -\frac{3}{4})\bigr\rbrack\bigr\rbrack$

Scope #

Exp(-x) grows without bound as x -> -Infinity

live ↗

In$\mathrm{Maximize}(\exp(-x), x)$Out$\bigl\lbrack∞, \bigl\lbrack\mathrm{Rule}(x, -∞)\bigr\rbrack\bigr\rbrack$

Sqrt(4 - x^2)'s domain is exactly [-2, 2] -- its vertex (x = 0) is the max

live ↗

In$\mathrm{Maximize}(\sqrt{4-x^2}, x)$Out$\bigl\lbrack2, \bigl\lbrack\mathrm{Rule}(x, 0)\bigr\rbrack\bigr\rbrack$

Possible issues #

Declines -- multivariate optimization is out of scope

live ↗

In$\mathrm{Maximize}(-(x^2+y^2), \bigl\lbrack x, y\bigr\rbrack)$Out$\mathrm{Maximize}(-(x^2+y^2), \bigl\lbrack x, y\bigr\rbrack)$

About

Examples	7
Checked against	Wolfram Language 6 agree1 differ
Crosswalk	Wolfram Language

See also: Minimize, MinValue, MaxValue, NMaximize, Solve, Limit