We need to optimize f(x, y, z) = x + 2y subject to the constraints g(x, y, z) = x + y + z = 8 and h(x, y, z) = y2 + z2 = 4. To find the possible extreme value points, we must use ∇f = λ∇g + μ∇h.

The Lagrangian is

with critical points where the partial derivatives vanish:

The second and third equations tell us , so that in the last equation we find

and from the fourth equation we get

So we have two critical points, and , which give respective extreme values of (maximum) and (minimum).

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step-by-step explanation: