Problems with two constraints Given a differentiable function w = f(x; y, z), the goal is to find its maximum and minimum values subject to the constraints g(x, y, z) = 0 and h(x, y, z) = 0, where g and h are also differentiable.
a. Imagine a level surface of the function f and the constraint surfaces g(x, y, z) = 0 and h(x, y, z) = 0. Note that g and h intersect (in general) in a curve C on which maximum and minimum values of f must be found. Explain why ▿g and ▿h are orthogonal to their respective surfaces.
b. Explain why ▿f lies in the plane formed by ▿g and ▿h at a point of C where f has a maximum or minimum value.
c. Explain why part (b) implies that ▿f = λ▿g + μ▿h at a point of C where f has a maximum or minimum value, where λ and μ. (the Lagrange multipliers) are real numbers.
d. Conclude from part (c) that the equations that must be solved for maximum or minimum values of f subject to two constraints are ▿f = λ▿g + μ▿h, g(x, y, z) = 0 and h(x, y, z) = 0.
Want to see the full answer?
Check out a sample textbook solutionChapter 15 Solutions
Calculus: Early Transcendentals (3rd Edition)
Additional Math Textbook Solutions
Glencoe Math Accelerated, Student Edition
Calculus, Single Variable: Early Transcendentals (3rd Edition)
Thomas' Calculus: Early Transcendentals (14th Edition)
Calculus & Its Applications (14th Edition)
Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (4th Edition)
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage LearningCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage