Use lagrange multipliers to find the maximum and minimum values of the function subject to the given constraints. (if a value does not exist, enter none.) f(x, y,z) = 3x – y – 3z; x + y – z = 0, x2 + 2z2 = 1

The Lagrangian is

with critical points where its partial derivatives vanish:

Since we know , we have

So there are two critical points, and , which give a minimum value of and a maximum value of , respectively.

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step-by-step explanation: