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.


Find the volume of the largest rectangular parallelepiped that can be inscribed in the ellipsoid 𝑥𝑎+𝑦𝑏+𝑧𝑐=1.

  • A8𝑎𝑏𝑐3
  • B𝑎𝑏𝑐33
  • C6𝑎𝑏𝑐3
  • D8𝑎𝑏𝑐33
  • E8𝑎𝑏𝑐


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

  • A𝑥=5, 𝑦=3, 𝑧=2
  • B𝑥=103, 𝑦=5, 𝑧=53
  • C𝑥=5, 𝑦=103, 𝑧=53
  • D𝑥=4, 𝑦=2, 𝑧=4
  • E𝑥=4, 𝑦=4, 𝑧=2

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