Worksheet: Lagrange Multipliers

In this worksheet, we will practice optimizing and finding the extrema of a multivariable function subject to a given constraint using Lagrange multipliers.

Q1:

Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid 𝑥 𝑎 + 𝑦 𝑏 + 𝑧 𝑐 = 1 .

  • A 6 𝑎 𝑏 𝑐 3
  • B 8 𝑎 𝑏 𝑐
  • C 𝑎 𝑏 𝑐 3 3
  • D 8 𝑎 𝑏 𝑐 3 3
  • E 8 𝑎 𝑏 𝑐 3

Q2:

Find the three positive numbers 𝑥 , 𝑦 , and 𝑧 whose sum is 10 such that 𝑥 𝑦 𝑧 is a maximum.

  • A 𝑥 = 5 , 𝑦 = 3 , 𝑧 = 2
  • B 𝑥 = 5 , 𝑦 = 1 0 3 , 𝑧 = 5 3
  • C 𝑥 = 4 , 𝑦 = 2 , 𝑧 = 4
  • D 𝑥 = 1 0 3 , 𝑦 = 5 , 𝑧 = 5 3
  • E 𝑥 = 4 , 𝑦 = 4 , 𝑧 = 2

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