Question

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

Answer 1

The points that maximize the product xy on the circle x²+y²=18 are (3,3) and (-3,-3)

Explanation:

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:

`\\abla f (x,y) = \lambda \, \\abla g (x,y)`

Where
`\lambda` is a constant. This means that there exists a constant

• 
  `f_x(x,y) = \lambda \, g_x(x,y)`
• 
  `f_y(x,y) = \lambda \, g_y(x,y)`

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:

• 
  `f_x(x,y) = 2x`
• 
  `f_y(x,y) = 2y`
• 
  `g_x(x,y) = y`
• 
  `g_y(x,y) = x`

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:

• 
  `2x = \lambda \, y`
• 
  `2y = \lambda \, x`
• 
  `x^2+y^2 = 18`

Since
`2x = \lambda \, y` , then
`x = (\lambda)/(2) \, y` . If we replace the value of x in the other equation, we obtain that

`2y = \lambda \, x = \lambda \, ((\lambda)/(2) \, y) = (\lambda^2)/(2) y`

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

`2x = ^+_- \, 2y`

`2y = ^+_- \, 2x`

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.