Example

]> 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 $\frac{1}{2}$ .

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