Find the maximum value of f(x,y,z) = x^2 + xy + y^2 + yz + z^2 on the unit sphere.
I don't know whether I should parametrize the unit sphere and use a composition of functions, or go straight into Lagrange multipliers. Either way, I don't know how to solve the resulting equations.
So you have to solve the following four equations
(1) 2x + y = 2λx
(2) x + 2y + z = 2λy
(3) y + 2z = 2λz
(4) x^2 + y^2 + z^2 = 1
Is there a usual way to solve a system like this?
What I did was solve (1) for y and plugged that y into (3). That gave me λ=1, which then gave me y=0, x = ± sqr(1/2), and z = ∓ sqr(1/2). However, this does not give the maximum value of f. Does this just mean that there are other points (x,y,z,λ) that satisfy these equations? How do you find all of them?