]> Example
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## Example

In our example we seek the maximum of $F$ : $F = x y$ , 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 $( 2 a x , 2 b y )$ .

The determinant of the matrix whose columns are these vectors is $2 b y 2 − 2 a x 2$ , 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 $1 2$ .

Since there are two $x$ values that obey this condition (namely, plus or minus the square root of $1 2 a$ , and similarly two $y$ values (plus or minus the square root of $1 2 b$ ), 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 $1 a b$ , and two minima with minus this value, when the roots for $x$ and $y$ have opposite sign.

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