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.

$\mathrm{extrema}\left(a{x}^2+bx+c\,\varnothing \,x\right)$

$\left\{-\frac{b^2}{4a}+c\right\}$

$\mathrm{extrema}\left(axyz\,x^2+y^2+z^2=1\,\left\{x\,y\,z\right\}\right)$

$\left\{\max \left(0\,-\frac{\sqrt{3}a}{9}\,\frac{\sqrt{3}a}{9}\right)\,\min \left(0\,-\frac{\sqrt{3}a}{9}\,\frac{\sqrt{3}a}{9}\right)\right\}$

$f≔{\left(x^2+y^2\right)}^{\frac{1}{2}}-z\;$$\mathrm{g1}≔x^2+y^2-16\;$$\mathrm{g2}≔x+y+z=10$

$f≔\sqrt{x^2+y^2}-z$

$\mathrm{g1}≔x^2+y^2-16$

$\mathrm{g2}≔x+y+z=10$

$\mathrm{extrema}\left(f\,\left\{\mathrm{g1}\,\mathrm{g2}\right\}\,\left\{x\,y\,z\right\}\,'s'\right)$

$\left\{-6-4\sqrt{2}\,-6+4\sqrt{2}\right\}$

$s$

$\left\{\left\{x=-2\sqrt{2}\,y=-2\sqrt{2}\,z=4\sqrt{2}+10\right\}\,\left\{x=2\sqrt{2}\,y=2\sqrt{2}\,z=-4\sqrt{2}+10\right\}\right\}$