Consider the function f(x, y, z) = x 3y + e yz . (a) Use Lagrange multipliers to write out a system of equations to solve for optimizing f subject to the constraint x 2 + y 2 = z 2 + 1. Call this System A. Do NOT attempt to solve your system. (b) Do the same thing for f subject to the constraint 3x 2 + 2y 2 + z 2 = 1. Call this System B. Again, do NOT attempt to solve your system
Consider the function f(x, y, z) = x 3y + e yz .
(a) Use Lagrange multipliers to write out a system of equations to solve for optimizing f subject to the constraint x 2 + y 2 = z 2 + 1. Call this System A. Do NOT attempt to solve your system.
(b) Do the same thing for f subject to the constraint 3x 2 + 2y 2 + z 2 = 1. Call this System B. Again, do NOT attempt to solve your system.
(c) Systems of n equations with n variables don’t always have solutions, but in this case we can guarantee that one of the systems you’ve written above must have a solution. Which is it and why? Hints: What points are your systems trying to find? What theorems do we have related to finding such points?
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