Use Lagrange multipliers to find all the points on the circle x^2 + y^2 = 18 at which the product xy is a maximum.

We have the function f(x,y) = xy, and we want to find its maximum with the restriction g(x,y) = 18; with g(x,y) = x²+y². The Lagrange multiplier theorem stays that a point (x,y) that is the maximum of f with the given restriction should fulfill the following:

[tex] \nabla f (x,y) = \lambda \, \nabla g (x,y) [/tex]

Where [tex] \lambda [/tex] is a constant. This means that there exists a constant

[tex] f_x(x,y) = \lambda \, g_x(x,y) [/tex]
[tex] f_y(x,y) = \lambda \, g_y(x,y) [/tex]

Where, for a differentiable function h, h_x and h_y are the partial derivates of h respect to the variables x and y respectively. The partial derivate, for example with respect to the variable x, is obtained by derivating the function thinking the variable y as a constant.

With this in mind lets compute the partial derivates of f and g:

[tex] f_x(x,y) = 2x [/tex]
[tex] f_y(x,y) = 2y [/tex]
[tex] g_x(x,y) = y [/tex]
[tex] g_y(x,y) = x[/tex]

So, if we replace each partial derivate by its formula in the relations we had before, and we add the restriction g(x,y) = 18, we obtain the following 3 conditions:

[tex] 2x = \lambda \, y [/tex]
[tex] 2y = \lambda \, x [/tex]
[tex] x^2+y^2 = 18 [/tex]

Since [tex] 2x = \lambda \, y [/tex] , then [tex] x = \frac{\lambda}{2} \, y [/tex] . If we replace the value of x in the other equation, we obtain that

[tex]2y = \lambda \, x = \lambda \, (\frac{\lambda}{2} \, y) = \frac{\lambda^2}{2} y[/tex]

This means that [tex] \lambda^2 = 4[/tex] , thus [tex] \lambda = 2 [/tex] or [tex] \lambda = -2 [/tex] . We can translate both equations therefore as:

[tex] 2x = ^+_- \, 2y [/tex]

[tex] 2y = ^+_- \, 2x [/tex]

Thus, y = x, or y = -x. In order for xy to be positive (and hence, have a chance to be a maximum), we will only care about x=y.

Lets replace y with x in the restriction given by gi:

g(x,x) = 18

x²+x² = 2x² = 18

x² = 9

x = 3 or x = -3

Therefore, the candidates for maximum for f with the restriction g(x,y) = 18 are (3,3) and (-3,-3). In both cases f(x,y) = 3*3 = (-3)*(-3) = 9. As a result, both points maximize the product xy on the circle.