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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 2by2-2ax2, so that our extremum condition, that this determinant is 0, becomes by2 = ax2. If we apply the condition G = 0, we see that each of by2 and ax2 must be 1/2. Since there are two x values that obey this condition (namely,
plus or minus the square root of |
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,
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.