Consider the problem of minimizing the function f(x, y) = x on the curve y² + x² - x³ = 0 (a piriform). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is f(0, 0) = 0 but the Lagrange condition Vf(0, 0) = AVg(0, 0) is not satis- fied for any value of A. (c) Explain why Lagrange multipliers fail to find the mini- mum value in this case. implicitly defined
Consider the problem of minimizing the function f(x, y) = x on the curve y² + x² - x³ = 0 (a piriform). (a) Try using Lagrange multipliers to solve the problem. (b) Show that the minimum value is f(0, 0) = 0 but the Lagrange condition Vf(0, 0) = AVg(0, 0) is not satis- fied for any value of A. (c) Explain why Lagrange multipliers fail to find the mini- mum value in this case. implicitly defined
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
25
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CHAPTER 14 Partial Derivatives
25. Consider the problem of minimizing the function
f(x, y) = x on the curve y² + x4 - x³ = 0 (a piriform).
(a) Try using Lagrange multipliers to solve the problem.
(b) Show that the minimum value is f(0, 0) = 0 but the
Lagrange condition Vf(0, 0) = AVg(0, 0) is not satis-
fied for any value of A.
(c) Explain why Lagrange multipliers fail to find the mini-
mum value in this case.
CAS 26. (a)
3
3
If your computer algebra system plots implicitly defined
curves, use it to estimate the minimum and maximum
values of f(x, y) = x³ + y³ + 3xy subject to the con-
straint (x - 3)² + (y - 3)2 = 9 by graphical methods.
(b) Solve the problem in part (a) with the aid of Lagrange
multipliers. Use your CAS to solve the equations numer-
ically. Compare your answers with those in part (a).
27. The total production P of a certain product depends on
the amount L of labor used and the amount K of capital
investment. In Sections 14.1 and 14.3 we discussed how the
Cobb-Douglas model P = bLºK¹-a follows from certain
economic assumptions, where b and a are positive constants
and a < 1. If the cost of a unit of labor is m and the cost
of a unit of capital is n, and the company can spend only
p dollars as its total budget, then maximizing the produc-
tion P is subject to the constraint mL + nK = p. Show
that the maximum production occurs when
von
L =
ap
m
and K =
(1 - a)p
n
28. Referring to Exercise 27, we now suppose that the pro-
duction is fixed at bL"K¹-a "=Q, where Q is a constant.
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