Solve the optimization problem. hint [see example 2.] maximize p = xyz with x + z = 24 and y + z = 24, and x, y, and z ≥ 0. p =

Let's use the method of Lagrange multipliers. We have the Lagrangian

with partial derivatives (set equal to 0)

From the last two equations, we have

so that

Then

If , then and , so one critical point occurs at (0, 0, 24) with a value of .

If , then [te]y=2z[/tex] and , making and giving another critical point at (16, 16, 8) with a value of .

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step-by-step explanation: