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.