HELP!!!! Using LaGrange multipliers; Find the extrema of f(x,y,z) = x+2y-3z subject to z = 4x^2+y^2, then use the discriminant to determine whether it is a max or a min.

Question

anonymous · Answer

What have you tried so far? 
1. Write out the Lagrange Equations
2. Solve for lambda in terms of x and y
3. solve for x and y using the constraint
4. Calculate the critical values.

anonymous · Answer

Like this?$$<8x,2y,z>=\lambda<1,2,-3>$$

anonymous · Answer

The Lagrange multiplier is on the wrong side.

anonymous · Answer

This?$$<1,2,-3>=\lambda<8x,2y,z>$$

anonymous · Answer

Close, what is the derivative of z? It's one. 
Now it's only a matter of setting up systems of linear equations. And substituting back into the constraint equation.

anonymous · Answer

So would it be.... <1,2,-3>=lambda<8x,2y,1> E1: 1=lambda*8x => x=1/8*lambda E2: 2=lambda*2y => y=1/lambda E3: -3=lambda*1 => lambda=-3 Right?

ganeshie8 · Answer

you should get :

$\large \langle1, 2, -3 \rangle =   \lambda  \langle 8x, 2y, -1\rangle$

ganeshie8 · Answer

that gives : $\large \lambda =  3$
you can find out x and y values easily