Need help in finding extreme values

[haruspex]

Given the equation x^2+y^2+3z^2-2xy-2yz=2.Find the local extrema for the function z=f(x,y).What do I do with the equation?

What do you get when you take the partial derivatives?

 

[aruwin]

What do you get when you take the partial derivatives?

Is this ok?
Partial derivative in respect to x is 2x-2y and in respect to y is 4y-2x-2z.
What do I do with z??

 

[HallsofIvy]

Thanks for all your explanations and patience.Yes,I have solved this question with your help and harupex too.By the wsy,I have a new question.Since it is related to finding local extrema too,I am just going to ask it here.
Given the equation x^2+y^2+3z^2-2xy-2yz=2.Find the local extrema for the function z=f(x,y).What do I do with the equation?

You want to differentiate z with respect to x and y and can do that using "implicit differentiation" just like you learned in Calculus I.

 

[haruspex]

Is this ok?
Partial derivative in respect to x is 2x-2y and in respect to y is 4y-2x-2z.
What do I do with z??

Partial derivative of z wrt x is ∂z/∂x, etc. Just treat that as a variable. You can differentiate it partially wrt x again, ∂²z/∂x², or wrt y: ∂²z/∂x∂y. (Btw, ∂²z/∂x∂y = ∂²z/∂y∂x)