Example

In our example we seek the maximum of F: F = xy, subject to the condition G = ax² + by² -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 , so that our extremum condition, that this determinant is 0, becomes . If we apply the condition G = 0, we see that each of

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