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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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