Lagrange Multipliers - unknown values

[maks4]

Homework Statement

Using Lagrange Multipliers, we are to find the maximum and minimum values of f(x,y) subject to the given constraint

Homework Equations

f(x,y,z) = x^2 - 2y + 2z^2, constraint: x^2 + y^2 + z^2 = 1

The Attempt at a Solution

grad f = lambda*grad g
(2x, -2, 4z) = lambda(2x, 2y, 2z)
therefore: 2x = lambda2x
-2 = lambda2y
4z = lambda2z

Now i can see how x and z can both equal 0, and how lambda can equal 1, and how y eventually equals -1 and f(0,-1,0)=2, yet in the solutions there is another part to it where it says
OR, Lambda = 2, y = -1/2, x = 0, z= +- sqrt(3)/2. <---i do not have any idea how these values came to be, no idea at all. Any help would be appreciated, thanks.

 

[Office_Shredder]

If

$$4z = 2z \lambda$$ then you get that either z=0, or if z is non-zero, it must be lambda=2. Then you just work from there - y comes easily next, followed up by x, and then using the constraint to solve for what z actually is

 

[maks4]

^^I see it now then, thank you.