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extrema

find relative extrema of an expression

 Calling Sequence extrema(expr, constraints) extrema(expr, constraints, vars) extrema(expr, constraints, vars, 's')

Parameters

 expr - expression whose extrema are to be found constraints - constraint or set of constraints vars - variable or set of variables s - unevaluated name

Description

 • The extrema function can be used to find extreme values of a multivariate expression with zero or more constraints. Candidates for extreme value points can also be returned. The extrema are returned as a set, and the candidates are returned as a set of sets of equations in the appropriate variables.
 • expr must be an algebraic expression. The constraints may be specified as either expressions or equations. When a constraint is given as an expression, it is understood that constraint = 0. If no constraints are to be given, then the empty set {} is used in the parameter list. If vars is not given then all name indeterminates in the expr and constraints are used. vars must be specified if the fourth parameter s is given. The candidates for the extreme value points are returned in s.
 • When the candidates cannot be expressed in closed form, s will contain the system of equations which when solved will produce these candidates.
 • This function employs the method of Lagrange multipliers.

Examples

 > $\mathrm{extrema}\left(a{x}^{2}+bx+c,\left\{\right\},x\right)$
 $\left\{\frac{{1}}{{4}}{}\frac{{4}{}{a}{}{c}{-}{{b}}^{{2}}}{{a}}\right\}$ (1)
 > $\mathrm{extrema}\left(axyz,{x}^{2}+{y}^{2}+{z}^{2}=1,\left\{x,y,z\right\}\right)$
 $\left\{{\mathrm{max}}{}\left({0}{,}{-}\frac{{1}}{{9}}{}\sqrt{{3}}{}{a}{,}\frac{{1}}{{9}}{}\sqrt{{3}}{}{a}\right){,}{\mathrm{min}}{}\left({0}{,}{-}\frac{{1}}{{9}}{}\sqrt{{3}}{}{a}{,}\frac{{1}}{{9}}{}\sqrt{{3}}{}{a}\right)\right\}$ (2)
 > $f≔{\left({x}^{2}+{y}^{2}\right)}^{\frac{1}{2}}-z;$$\mathrm{g1}≔{x}^{2}+{y}^{2}-16;$$\mathrm{g2}≔x+y+z=10$
 ${f}{:=}\sqrt{{{x}}^{{2}}{+}{{y}}^{{2}}}{-}{z}$
 ${\mathrm{g1}}{:=}{{x}}^{{2}}{+}{{y}}^{{2}}{-}{16}$
 ${\mathrm{g2}}{:=}{x}{+}{y}{+}{z}{=}{10}$ (3)
 > $\mathrm{extrema}\left(f,\left\{\mathrm{g1},\mathrm{g2}\right\},\left\{x,y,z\right\},'s'\right)$
 $\left\{{-}{6}{-}{4}{}\sqrt{{2}}{,}{-}{6}{+}{4}{}\sqrt{{2}}\right\}$ (4)
 > $s$
 $\left\{\left\{{x}{=}{-}{2}{}\sqrt{{2}}{,}{y}{=}{-}{2}{}\sqrt{{2}}{,}{z}{=}{4}{}\sqrt{{2}}{+}{10}\right\}{,}\left\{{x}{=}{2}{}\sqrt{{2}}{,}{y}{=}{2}{}\sqrt{{2}}{,}{z}{=}{-}{4}{}\sqrt{{2}}{+}{10}\right\}\right\}$ (5)