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
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