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**snaes** Use Lagrange multipliers to find the max and min values of the function subject to the given constraints.

I get multiple answers for $\displaystyle \lambda$, which is my problem.

function:

$\displaystyle f(x,y,z)=x+2y$

subject to:

$\displaystyle x+y+z=1 $ and $\displaystyle y^2+z^2=4$

$\displaystyle \nabla f <1,2,0> = \lambda<1,1,1>$

$\displaystyle \nabla f <1,2,0> = \lambda<0,2y,2z>$

Now I set them equal to solve for $\displaystyle \lambda$ using all of the possible combinations, here is where my answers get messed up. They are all different, and $\displaystyle 1= \lambda 0$ makes no sense.

$\displaystyle 1= \lambda$

$\displaystyle 2= \lambda$

$\displaystyle 0= \lambda$

$\displaystyle 1= \lambda 0$

$\displaystyle 2= \lambda 2y$

$\displaystyle 0= \lambda 2z$

From here any solution i pick does not give me the solution of $\displaystyle 1,\sqrt{2},-\sqrt{2}$ as a max. Also $\displaystyle 1,-\sqrt{2},\sqrt{2}$ is the min.

Any help is appreciated as always.

Thanks!