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Constrained Extrema of Bivariate Functions

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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 $ x=0$, $ y=3$ is not possible) yields

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

Together with $ g = 0$, it follows now that $ (1,1)$ and $ (-1,-1)$ are the only solutions of these equations. Since a continuous function on a compact set has a minimum and a maximum, $ f$ is minimal at $ (1,1)$ and maximal at $ (-1,-1)$.

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  automatisch erstellt am 26.  1. 2017