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Back to Constrained Optimization: Lagrange Multipliers

Exercise 14

Avg difficulty 2.5

Apply the Method of Lagrange Multipliers to solve each of the following constrained optimization problems.

  1. Determine the absolute maximum and absolute minimum values of f(x,y)=(x−1)2+(y−2)2f(x,y) = (x-1)^2 + (y-2)^2 subject to the constraint that x2+y2=16x^2 + y^2 = 16.

  2. Determine the points on the sphere x2+y2+z2=4x^2 + y^2 + z^2 = 4 that are closest to and farthest from the point (3,1,−1)(3,1,-1). (As in the preceding exercise, you may find it simpler to work with the square of the distance formula, rather than the distance formula itself.)

  3. Find the absolute maximum and minimum of f(x,y,z)=x2+y2+z2f(x,y,z) = x^2 + y^2 + z^2 subject to the constraint that (x−3)2+(y+2)2+(z−5)2≤16(x-3)^2 + (y+2)^2 + (z-5)^2 \le 16. (Hint: here the constraint is a closed, bounded region. Use the boundary of that region for applying Lagrange Multipliers, but don't forget to also test any critical values of the function that lie in the interior of the region.)

Long, Keller & Schlicker, Active Calculus Multivariable, §11.11 Constrained Optimization: Lagrange Multipliers, Exercise 14

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Adapted from Active Calculus Multivariable © 2013–2026 Nicholas Long, Mitchel T. Keller, and Steven Schlicker. Licensed under CC BY-SA 4.0. Source: Active Calculus Multivariable. Imported at revision 692c03c.