Quiz 7 Optimization Problems with constraints: Lagrange Multipliers

Problem 1. Find the values of x,y and z that maximize the function f(x,y,z)=xyz, if x,y and z satisfy the relation x+2y+2z=108, and x, y and z are all non-negative. (This is basically the package problem of Quiz 6.)

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Lagrange multipliers: A Review

Complete solution

Problem 2) Does the function f(x,y,z)=xyz have a maximum value on the plane defined by the relation x+2y+2z=108?

A) Yes, at x=36, y=18, z=18.

B) No.

C) You can't tell.

Problem 3) Below is a plot of the gradient field of f(x,y)=3x ²+5y ², and a curve. Based on the plot, label the points on the curve where f has a local max or local min. (We have re-scaled the gradient vectors so that they are all unit vectors to make the plot clearer.)

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Complete solution

Problem 4) A current of 6 amperes branches into currents x, y, and z through resistors with resistences 2, 4,and 5 ohms as shown.

It is known that the current splits in such a way that the sum of the currents through the three resistors equals the initial current. The energy generated in each resistor is given by E=I²R, where I is the current in that resistor and R is the resistance. Use Lagrange multipliers to find the currents x, y, and z which will minimize the total energy generated. (It turns out that nature always splits the currents so that the total energy is minimized.)

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Complete solution

How to solve Max Min problems using Lagrange multipliers

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