Chapter 12.9, Problem 61E

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.