Home  18.013A  Chapter 14  Section 14.2 


In our example we seek the maximum of $F$ : $F=xy$ , subject to the condition $G=a{x}^{2}+b{y}^{2}1=0$ .
The gradient of $F$ is $(y,x)$ and the gradient of $G$ is $(2ax,2by)$ .
The determinant of the matrix whose columns are these vectors is $2b{y}^{2}2a{x}^{2}$ , so that our extremum condition, that this determinant is 0, becomes $b{y}^{2}=a{x}^{2}$ . If we apply the condition $G=0$ , we see that each of $b{y}^{2}$ and $a{x}^{2}$ must be $\frac{1}{2}$ .
Since there are two
$x$
values that obey this condition (namely, plus or minus the square root of
$\frac{1}{2a}$
, and similarly two
$y$
values (plus or minus the square root of
$\frac{1}{2b}$
), there are four solutions of the extremum condition. It is easy to see that there are two maxima, when the roots have the same sign for
$x$
and
$y$
, with value one half the square root of
$\frac{1}{ab}$
, and two minima with minus this value, when the roots for
$x$
and
$y$
have opposite sign.

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