7. f(x, y, z) = 2x + 2y + z; x² + y² + z² = 9
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
7
Transcribed Image Text:20
YA
0
29
м
and so μ² = 22, μ =
70
9.
⁹. f(x, y, z) = xy²z;
z = 1 = x + y = 1 ± 7/√29. The corresponding values of f are
14.8 EXERCISES
g(x, y) = 8. Estimate the maximum and minimum values
1. Pictured are a contour map of f and a curve with equation
off subject to the constraint that g(x, y)
reasoning.
= 8. Explain your
√/29 + 2( = √/29) + 3(1
=
Therefore the maximum value of f on the given curve is 3 + √29
g(x, y) = 8
60
50
40
30
20
10
x² + y² = 1
x² + y² = 10
X
3. f(x, y) = x² - y²;
4. f(x, y) = 3x + y;
5. f(x, y) = xy; 4x² + y² = 8
6. f(x,y) = xe”; x + y = 2
7. f(x, y, z) = 2x + 2y + z; x² + y² + z² = 9
8. f(x, y, z) = exy²;
2x² + y² + z² = 24
x² + y² + z² = 4
It
+
2
2. (a) Use a graphing calculator or computer to graph the
circle x² + y² = 1. On the same screen, graph several
curves of the form x² + y = c until you find two that
just touch the circle. What is the significance of the
values of c for these two curves?
± √29/2.
Then x =
± √29/2. Then x = 2/√√29, y = ±5/√29, and, from (20),
(b) Use Lagrange multipliers to find the extreme values of
f(x, y) = x² + y subject to the constraint x² + y² = 1.
Compare your answers with those in part (a).
D
3-14 Each of these extreme value problems has a solution with
both a maximum value and a minimum value. Use Lagrange
multipliers to find the extreme values of the function subject to
the given constraint.
01
10. f(x, y, z) = ln(x² + 1) + ln(y² + 1) + ln(z² + 1);
x² + y² + z² = 12
7
11. f(x, 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 + z² + t² = 1
14. f(x1, x2,...,xn) = x₁ + x₂ +
x₂ + ... + xn;
x² + x² + + x² = 1
x₁ +
...
=
29
= 3 ± √29
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
x² + y² subject to the constraint xy
f(x, y)
1 can be
solved using Lagrange multipliers, but f does not have a
maximum value with that constraint.
DE
=
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. qu
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
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
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 by
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) = exy, x² + 4y² ≤ 1
GENE
X
24. Consider the problem of maximizing the function
f(x, y) = 2x + 3y subject to the constraint √√x + √√y = 5.
(a) Try using Lagrange multipliers to solve the problem.
(b) Does f(25, 0) give a larger value than the one in part (a)?
(c) Solve the problem by graphing the constraint equation
and several level curves of f.
(d) Explain why the method of Lagrange multipliers fails to
solve the problem.
(e) What is the significance of f(9, 4)?
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