<s>[INST] Use Lagrange Multipliers to determine max and min Using Lagrange Multipliers, determine the maximum and minimum of the function
$f(x,y,z) = x + 2y$
subject to the constraints
$x + y + z = 1$ and $y^2 + z^2 = 4$:
Justify that the points you have found give the maximum and minimum of $f$.
So, 
$$
\nabla f  = (λ_1)\nabla g_1 + (λ_2)\nabla g_2
$$
I get to this point
$$
(1,2,0) = λ_1(1,1,1) + λ_2(0,2y,2z)
$$
Where do I go from here to find the critical points ect.
 [/INST] We can find the extreme values using one constraint only: $z = 1 - x - y$, so $y^2 + (1-x-y)^2 = 4$.
Thus: $g(x,y) = y^2 + 1 + x^2 + y^2 - 2x - 2y + 2xy - 4 = 0$.
So: $\nabla f = \lambda\nabla g$ gives:
$(1,2) = \lambda(2x-2+2y,4y-2+2x)$. Thus: 
$\dfrac{1}{2x-2+2y} = \dfrac{2}{4y-2+2x}$. Hence:
$4y-2+2x = 4x - 4 + 4y$. So $x = 1$, and $y^2 + y^2 = 4$. So: $y^2 = 2$, and $y = \pm \sqrt{2}$. Thus we easily see that extrema are:
$f_{min} = f(1,-\sqrt{2}) = 1 - 2\sqrt{2}$, and $f_{max} = f(1,\sqrt{2}) = 1 + 2\sqrt{2}$.
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