Use Lagarange multipliers to find the indicated extrema of f, subject to two constraints. In each case, assume that x,y and z are nonnegative.

Minimize: f(x,y,z)=x^2+y^2+z^2
Constraints: g(x,y,z) = x+2z=6,h(x,y,z)=x+y=12

Question

Use Lagarange multipliers to find the indicated extrema of f, subject to two constraints. In each case, assume that x,y and z are nonnegative. 

Minimize: f(x,y,z)=x^2+y^2+z^2
Constraints: g(x,y,z) = x+2z=6,h(x,y,z)=x+y=12

anonymous · Answer

after I get the derivatives and set f each to lambda deta g + mu deltah. I'm stuck. I'm not sure on what equation I should be plugging in... I'll post what I have so far up.

anonymous · Answer

Minimize: $$\Delta f(x,y,z)=<2x,2y,2z>$$ $$\lambda \Delta g(x,y,z) =<\lambda,2\lambda>$$ $$\mu Deltah(x,y,z)=<\mu,\mu>$$

anonymous · Answer

The last two are the derivatives(partials) of the constraints

anonymous · Answer

Now setting them equal to delta F I get this:

$$2x=\lambda+\mu$$
$$2y=\mu$$
$$2z=2\lambda$$

anonymous · Answer

From there I'm stuck.

anonymous · Answer

would i have to solve for x y and z or set one of the above equations equal to each other
?