5-20 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 5. f(x, y) = x² + xy + y² + y 6. f(x, y) = xy - 2x - 2y - x² - y² Gbna 7. f(x, y) = (x - y)(1 − xy) 8. f(x, y) = y(e¹ - 1) 9. f(x, y) = x² + y² + 2xy 10. f(x, y) = 2x² + 2x² - y² 11. f(x, y) = x³ - 3x + 3xy² 12. f(x, y) = x³ + y³ - 3x² - 3y² - 9x 13. f(x, y) = x - 2x² + y² - 3y 14. f(x, y) = y cos x 15. f(x, y) = e' cos y 16. f(x, y) = xye-(x² + y²)/2 17. f(x, y) = xy + ex 18. f(x, y) = (x² + y²)e¯* 19. f(x, y) = y² - 2y cos x, -1≤x≤7 20. f(x, y) = sin x sin y, -
5-20 Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. 5. f(x, y) = x² + xy + y² + y 6. f(x, y) = xy - 2x - 2y - x² - y² Gbna 7. f(x, y) = (x - y)(1 − xy) 8. f(x, y) = y(e¹ - 1) 9. f(x, y) = x² + y² + 2xy 10. f(x, y) = 2x² + 2x² - y² 11. f(x, y) = x³ - 3x + 3xy² 12. f(x, y) = x³ + y³ - 3x² - 3y² - 9x 13. f(x, y) = x - 2x² + y² - 3y 14. f(x, y) = y cos x 15. f(x, y) = e' cos y 16. f(x, y) = xye-(x² + y²)/2 17. f(x, y) = xy + ex 18. f(x, y) = (x² + y²)e¯* 19. f(x, y) = y² - 2y cos x, -1≤x≤7 20. f(x, y) = sin x sin y, -
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
13
Transcribed Image Text:968
CHAPTER 14 Partial Derivatives
4. f(x, y) = 3x - x³ - 2y² + y4
1.5.
y
6. f(x, y) = xy - 2x - 2y - x² - y²
-
7. f(x, y) = (x − y)(1 − xy)
8. f(x, y) = y(ex - 1)
9. f(x, y) = x² + y² + 2xy
1651
10. f(x, y) = 2x² + 2x² - y²
1.5
5-20 Find the local maximum and minimum values and saddle
point(s) of the function. If you have three-dimensional graphing
software, graph the function with a domain and viewpoint that
reveal all the important aspects of the function.
5. f(x, y) = x² + xy + y² + y
16. f(x, y) = xye
11. f(x, y) = x³ - 3x + 3xy²
294085dw
12. f(x, y) = x³ + y³ − 3x² − 3y² – 9x
13. f(x, y) = x² - 2x² + y³ - 3y
14. f(x, y) = y cos x
15. f(x, y) = e* cos y
‚—(x² + y²)/2
16X
17. f(x, y) = xy + e¯xy
18. ƒ(x, y) = (x² + y²)e¯x
19. f(x, y) = y² - 2y cos x, -1 ≤x≤7
20. f(x, y) = sin x sin y,
100
0 <1
anoisibong
-T<x<T, -π<y< π
+ 2 has an infinite
=
21. Show that f(x, y) = x² + 4y² - 4xy
number of critical points and that D 0 at each one. Then
show that f has a local (and absolute) minimum at each
critical point.
22. Show that f(x, y) = x²ye-²-² has maximum values at
(±1, 1/√2) and minimum values at (±1, -1/√2). Show
also that f has infinitely many other critical points and
D = 0 at each of them. Which of them give rise to maximum
values? Minimum values? Saddle points?
23-26 Use a graph or level curves or both to estimate the local
maximum and minimum values and saddle point(s) of the
function. Then use calculus to find these values precisely.
-2
23. f(x, y) = x² + y² + x-²y-²
24. f(x, y) = (x - y)e-x²-y²
25. f(x, y) = sin x + sin y + sin(x + y),
0≤x≤2T, 0≤ y ≤ 2T
v)
A
26. f(x, y) = sin x + sin y + cos(x + y),
0≤x≤ π/4, 0≤ y ≤ π/4
27-30 Use a graphing device as in Example 4 (or Newton's
method or solve numerically using a calculator or computer) to
find the critical points of f correct to three decimal places. Then
classify the critical points and find the highest or lowest points on
the graph, if any.
27. f(x, y) = x² + y² - 4x²y + 2y
28. f(x, y) = y6 - 2y + x² - y² + y
29. f(x, y) = x² + y³ - 3x² + y² + x - 2y + 1
30. f(x, y) = 20e-²-sin 3x cos 3y, |x|<1, |y|<1
31-38 Find the absolute maximum and minimum values of f on
the set D.
31. f(x, y) = x² + y² - 2x, D is the closed triangular region
with vertices (2, 0), (0, 2), and (0, -2)
32. f(x, y) = x + y - xy, D is the closed triangular region
with vertices (0, 0), (0, 2), and (4,0)
its massi
33. f(x, y) = x² + y² + x²y + 4,
M
D = {(x, y) | |x| ≤ 1, |y| ≤ 1}
34. f(x, y) = x² + xy + y² - 6y,
D = {(x, y) | -3 ≤x≤ 3,0 ≤ y ≤ 5}
35. f(x, y) = x² + 2y² - 2x - 4y + 1,
D = {(x, y) | 0 ≤ x ≤ 2,0 ≤ y ≤ 3}
!)szoqq2
$.10 10100
36. f(x, y) = xy², D = {(x, y) | x ≥ 0, y = 0,₁ x² + y² ≤ 3}
37. f(x, y) = 2x³ + y4, D = {(x, y) | x² + y² ≤ 1}
38. f(x, y) = x³ - 3x - y3 + 12y, D is the quadrilateral
whose vertices are (-2, 3), (2, 3), (2, 2), and (-2,-2)
PARDEX
FAR
39. For functions of one variable it is impossible for a continuous
function to have two local maxima and no local minimum.
But for functions of two variables such functions exist. Show
that the function
5/6
f(x, y) = (x² - 1)² - (x²y - x - 1)²
has only two critical points, but has local maxima at both of
them. Then use a computer to produce a graph with a care-
fully chosen domain and viewpoint to see how this is
possible.
souns oss is mudunum 10 muraiz
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