Solving Maximum & Minimum of f(x,y) with Lagrange Multipliers

[Benny]

Hi, I'm having trouble with the following question.

Q. Find the maximum and minimum of the function f(x,y) = x^2 + xy + y^2 on the circle x^2 + y^2 = 1.

I started off by writing:

Let g(x,y) = x^2 + y^2 then $$\nabla f = \lambda \nabla g,g\left( {x,y} \right) = 1$$

$$\Rightarrow 2x + y = 2\lambda x...(1)$$

$$2y + x = 2\lambda y...(2)$$

$$x^2 + y^2 = 2...(3)$$

I'm not sure how to solve this system of equations. I've got the impression that generally an explicit value for lamda is not needed. I'd normally start off by considering the possible cases.

Here, if x = 0 in (1) then from (2) I get y = 0 so that (x,y) = (0,0). However this contradicts (3) and it isn't on the circle so I ignore this point. But now I'm stuck. Which values of x or y should I try now? Any help would be good thanks.

 

[Galileo]

Add equations (1) and (2).

Is your constraint g(x,y)=1 or g(x,y)=2?

 

[HallsofIvy]

A fairly common way of handling equations like that, where you don't need to find $\lambda$ is to divide one equation by the other.
You get $\frac{2x+y}{2y+x}= \frac{x}{y}$ or y(2x+y)= x(2y+ x). That obviously gives x²= y² so that either y= x or y= -x and that, together with x²+ y²= 1 gives you the answer.

By the way, since we have the constraint x²+ y²= 1, f(x,y)= x²+ xy+ y² immediately reduces to
f(x,y)= 1+ xy on that circle. That makes the calculations a little easier.

 

[Benny]

Thanks for the help Galileo and HallsofIvy.

The constraint is g(x,y) = 1, I'll fix it up now.