17-20 Find the extreme values of f subject to both constraints. 17. f(x, y, z) = x + y + z; x² + z² = 2₁ x + y = 1 18. f(x, y, z) = z; x² + y² = z², x+y+z=24 19. f(x, y, z) = yz + xy; xy = 1, y² + z² = 1 2
17-20 Find the extreme values of f subject to both constraints. 17. f(x, y, z) = x + y + z; x² + z² = 2₁ x + y = 1 18. f(x, y, z) = z; x² + y² = z², x+y+z=24 19. f(x, y, z) = yz + xy; xy = 1, y² + z² = 1 2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
19
Transcribed Image Text:8. Explain your
40
x
graph the
graph several
ind two that
nce of the
me values of
x² + y² = 1.
a).
solution with
Lagrange
n subject to
y, z) = x² + y² + z²; x² + y² + z² = 1
12. f(x, y, z) = x² + y² + z²; x² + y² + z² = 1
13. f(x, y, z, t) = x+y+z+t; x² + y² + 2² + ₁² = 1
14. f(x1,x2,..., Xn) X1
= x₁ +
x² + x² +
+ x² = 1
...
x₂ +
+ x₂ + ... + xn's
1 can be
15. The method of Lagrange multipliers assumes that the
extreme values exist, but that is not always the case.
Show that the problem of finding the minimum value of
f(x, y) = x² + y² subject to the constraint xy
solved using Lagrange multipliers, but f does not have a
maximum value with that constraint.
1
16. Find the minimum value of f(x, y, z) = x² + 2y² + 3z²
subject to the constraint x + 2y + 3z = 10. Show that f
has no maximum value with this constraint.
17-20 Find the extreme values of f subject to both constraints.
17. f(x, y, z) = x + y + z; x² + z² = 2, x + y = 1
x+y+z= 24
18. f(x, y, z) = z; x² + y² = z²,
19. f(x, y, z) = yz + xy; xy = 1, y² + z² = 1
20. f(x, y, z) = x² + y² + z²; x - y = 1, y² - z² = 1
21-23 Find the extreme values of f on the region described
the inequality.
21. f(x, y) = x² + y² + 4x - 4y, x² + y² ≤ 9
22. f(x, y) = 2x² + 3y² - 4x - 5, x² + y² ≤ 16
23. f(x, y) = e, x² + 4y² ≤ 1
24. Consider the problem of maximizing the function
f(x, y) = 2x + 3y subject to the constraint
√x
+ V
(a) Try using Lagrange multipliers to solve the pro
(h) Does f(25, 0) give a larger value than the one in
hy graphing the constraint e
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