Mathematik-Online lexicon: Constrained Extrema of Bivariate Functions

	[home] [lexicon] [problems] [tests] [courses] [auxiliaries] [notes] [staff]
	Mathematik-Online lexicon:
	Constrained Extrema of Bivariate Functions

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

For the constraint minimization problem

$\displaystyle f(x,y)=(x-3)(y-3)\rightarrow \min, \quad g(x,y)= x^2+y^2-2=0 \,,$

the Lagrange condition $\operatorname{grad}f \parallel \operatorname{grad} g$ is

$\displaystyle (y-3,x-3)=\lambda(2x,2y) \,.$

Elimination of $\lambda$ (note that

is not possible) yields

$\displaystyle (x-3)-\frac{y-3}{2x}2y=0 \quad \Leftrightarrow \quad (x-y)(x+y-3)=0 \,.$

Together with

, it follows now that

and

are the only solutions of these equations. Since a continuous function on a compact set has a minimum and a maximum,

is minimal at

and maximal at

automatisch erstellt am 26. 1. 2017