3.5 PROBLEMS In Problems I through 10, state whether the given function at tains a maximum value or a minimum value (or both) on the given interval (Suggestion: Begin by sketching a graph of the function/ 1. f(x) 1-x; -1.1) $f(x)=x: (1.1) 5. f(x)=x-21: (1.4) 7. f(x)+1; -1.1] & f(x)= (-00,00) 1 9. f(x) = x(1-x) 10. f(x) = 1. ZOTO this lo S=0.4 Calcula factor like a horizontal line (as in Fig. 3.5.1 in this manner? ke the c zontal ta Ordinate of esponding point with m alue attainet 1 2 f(x)=2x+1: (1.1) 4. 700) = (0.1) 6. f(x)=5-²: 1-1,2) [2,3] (0,1) *(1-x) In Problems 11 through 40, find the maximum and minimum values attained by the given function on the indicated closed interval 11. f(x) = 3x-2; [-2.3] 12 f(x)=4-3x: [-1.5] 13, h(x)=4-x²¹: [1.3] 14. f(x)=x²+3; [0.5] 1 g(x)=(x-1): [-1.4] 16. h(x)=x² + 4x +7: [-3.0] 17. f(x)=x²-3x: [-2,4] 18 g(x)=2r³-9x² + 12x: (0.4] 19. h(x)=x+ [1.4] 16 [1.3] 21. f(x)=3-2x; [-1.1] 22. f(x)=x²-4x+3; [0,2] 23. f(x)=5-12x-9x²; [-1,1] 24. f(x)=2x² - 4x +7; [0.2] 25. f(x)=x²-3x²-9x+5; [-2.4] ed interval (a. 20. f(x)=x² + = (if any) of f. minimum nor a ka ossibility that fis ferentiable there xtremum in (a, b) m, but only one ca local minimum, and in (a.b). 28. f(x)-(2x-3): [2.2] 29. fux)-5+17-3x: (1.5) 30. f(x)=x+11+x-11: (-2,23 31. f(x)=50-105x²+72x: (0.1) 32. fux)=2x+ (1.4) 33, 70x)= (0.3) 26. f(x)=r³+x: [-1.2] 27. f(x)=3x³-5x³; [-2,2] *+1 34. f(x)=110,3) 1-x 35. f(x)= 56. fox)-2- -1.8) 37. f(x)=x√1-¹: [-1.1] 38. f(x)=x√/4-¹ (0.21 39. f(x)=x2-x) (1.3) 40. f(x)=x2-x² (0.4) 41. Suppose that fix) Ax+ B is a linear function and that A 0. Explain why the maximum and minimum values of fon a closed interval (a, b) must occur at the endpoints of the interval. (-2,5) 42. Suppose that is continuous on [a, b] and differentiable on (a, b) and that f'(x) is never zero at any point of (a, b). Explain why the maximum and minimum values of f must occur at the endpoints of the interval (a, b). 43. Explain why every real number is a critical point of the greatest integer function f(x)=[x]. 44. Prove that every quadratic function f(x) = ax²+bx+c (0) has exactly one critical point on the real line. 45. Explain why the cubic polynomial function f(x) = ax + bx² +cx+d (a*0) can have either two, one, or no critical points on the real line. Produce examples that illustrate each of the three cases. 46. Define f(x) to be the distance from x to the nearest integ What are the critical points of f
3.5 PROBLEMS In Problems I through 10, state whether the given function at tains a maximum value or a minimum value (or both) on the given interval (Suggestion: Begin by sketching a graph of the function/ 1. f(x) 1-x; -1.1) $f(x)=x: (1.1) 5. f(x)=x-21: (1.4) 7. f(x)+1; -1.1] & f(x)= (-00,00) 1 9. f(x) = x(1-x) 10. f(x) = 1. ZOTO this lo S=0.4 Calcula factor like a horizontal line (as in Fig. 3.5.1 in this manner? ke the c zontal ta Ordinate of esponding point with m alue attainet 1 2 f(x)=2x+1: (1.1) 4. 700) = (0.1) 6. f(x)=5-²: 1-1,2) [2,3] (0,1) *(1-x) In Problems 11 through 40, find the maximum and minimum values attained by the given function on the indicated closed interval 11. f(x) = 3x-2; [-2.3] 12 f(x)=4-3x: [-1.5] 13, h(x)=4-x²¹: [1.3] 14. f(x)=x²+3; [0.5] 1 g(x)=(x-1): [-1.4] 16. h(x)=x² + 4x +7: [-3.0] 17. f(x)=x²-3x: [-2,4] 18 g(x)=2r³-9x² + 12x: (0.4] 19. h(x)=x+ [1.4] 16 [1.3] 21. f(x)=3-2x; [-1.1] 22. f(x)=x²-4x+3; [0,2] 23. f(x)=5-12x-9x²; [-1,1] 24. f(x)=2x² - 4x +7; [0.2] 25. f(x)=x²-3x²-9x+5; [-2.4] ed interval (a. 20. f(x)=x² + = (if any) of f. minimum nor a ka ossibility that fis ferentiable there xtremum in (a, b) m, but only one ca local minimum, and in (a.b). 28. f(x)-(2x-3): [2.2] 29. fux)-5+17-3x: (1.5) 30. f(x)=x+11+x-11: (-2,23 31. f(x)=50-105x²+72x: (0.1) 32. fux)=2x+ (1.4) 33, 70x)= (0.3) 26. f(x)=r³+x: [-1.2] 27. f(x)=3x³-5x³; [-2,2] *+1 34. f(x)=110,3) 1-x 35. f(x)= 56. fox)-2- -1.8) 37. f(x)=x√1-¹: [-1.1] 38. f(x)=x√/4-¹ (0.21 39. f(x)=x2-x) (1.3) 40. f(x)=x2-x² (0.4) 41. Suppose that fix) Ax+ B is a linear function and that A 0. Explain why the maximum and minimum values of fon a closed interval (a, b) must occur at the endpoints of the interval. (-2,5) 42. Suppose that is continuous on [a, b] and differentiable on (a, b) and that f'(x) is never zero at any point of (a, b). Explain why the maximum and minimum values of f must occur at the endpoints of the interval (a, b). 43. Explain why every real number is a critical point of the greatest integer function f(x)=[x]. 44. Prove that every quadratic function f(x) = ax²+bx+c (0) has exactly one critical point on the real line. 45. Explain why the cubic polynomial function f(x) = ax + bx² +cx+d (a*0) can have either two, one, or no critical points on the real line. Produce examples that illustrate each of the three cases. 46. Define f(x) to be the distance from x to the nearest integ What are the critical points of f
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
Q21, Q26 and Q35 needed
These are easy questions please solve both in the order to get positive feedback
![3.5 PROBLEMS
In Problems I through 10, state whether the given function at
tains a maximum value or a minimum value (or both) on the
given interval (Suggestion: Begin by sketching a graph of the
function/
1. f(x) 1-x; -1.1)
$f(x)=x: (-1.1)
5. f(x)=x-21: (1.4)
7. f(x)=x+1; -1.1]
1
& f(x)= (-00,00)
9. f(x) = x(1-x)
m
ZETO
this low
5=0.4
Calcubo
10. f(x)
=
ke the one
zontal ta
Ordinate of
esponding
point with
alue attainet
factor
like a horizontal line (as in Fig. 3.5.1
in this manner?
[2,3]
(0,1)
2 f(x)=2x+1: [-1.1)
1
4. f(x)=√ (0.1)
6. f(x)=5-²: 1-1,2)
*(1-x)
In Problems 11 through 40, find the maximum and minimum
values attained by the given function on the indicated closed
interval
11. f(x) = 3x-2;
[-2.3]
12 f(x)=4-3x: [-1.5]
13. h(x)=4-x²¹: [1.3]
14. f(x)=x²+3; [0.5]
28. f(x)-(2x-3): (1.2)
29. f(x)-5+17-3x: (1.5)
30. f(x)=x+11+x-11: (-2,23
31. f(x)=50-105x²+72x: 10.13
32. fux)=2x+ (1.4)
33700)= 10.31
34. f(x)=110,3)
35. f(x)=(-2,5)
*+1
56. fox)-2- (-1.5
37. f(x)=x√1-¹ (-11)
38. f(x)=x√/4-¹ (0.2)
39. f(x)=x2-x)
40. f(x)=x2-x¹²,
1 g(x)=(x-1): [-1.4]
16. h(x)=x² + 4x +7: [-3.0]
17. f(x)=x²-3x: [-2,4]
18 g(x)=2r³-9x² + 12x: (0.4]
19. h(x)=x+ [1.4]
16
[1.3]
21. f(x)=3-2x; [-1.1]
23. f(x)=5-12x-9x²; [-1,1]
22. f(x)=x²-4x+3; [0,2]
24. f(x)=2x² - 4x +7; [0.2]
25. f(x)=x²-3x²-9x+5; [-2.4]
ed interval (a. 20. f(x)=x²+ =
(if any) of f.
minimum nor a ka
ossibility that fs
ferentiable there
xtremum in (a, b)
m, but only one cri
local minimum, and
in (a, b).
26. f(x)=³+x: [-1.2]
27. f(x)=3x³-5x³; [-2.2]
(1.3)
(0.4)
41. Suppose that fix) Ax+B is a linear function and that
A 0. Explain why the maximum and minimum values of
fon a closed interval (a, b) must occur at the endpoints of
the interval.
42. Suppose that is continuous on [a, b] and differentiable
on (a, b) and that f'(x) is never zero at any point of (a, b).
Explain why the maximum and minimum values of f must
occur at the endpoints of the interval (a, b).
43. Explain why every real number is a critical point of the
greatest integer function f(x)=[x].
44. Prove that every quadratic function
f(x) = ax²+bx+c (20)
has exactly one critical point on the real line.
45. Explain why the cubic polynomial function
f(x) = ax + bx² +cx+d (a*0)
can have either two, one, or no critical points on the real line.
Produce examples that illustrate each of the three cases.
46. Define f(x) to be the distance from x to the nearest integ
What are the critical points of f](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffbfb5a78-2b03-4613-890c-0754521d637e%2F39b2d970-c1df-4c74-9df4-b9010a9f1a2d%2Fjrcg65g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3.5 PROBLEMS
In Problems I through 10, state whether the given function at
tains a maximum value or a minimum value (or both) on the
given interval (Suggestion: Begin by sketching a graph of the
function/
1. f(x) 1-x; -1.1)
$f(x)=x: (-1.1)
5. f(x)=x-21: (1.4)
7. f(x)=x+1; -1.1]
1
& f(x)= (-00,00)
9. f(x) = x(1-x)
m
ZETO
this low
5=0.4
Calcubo
10. f(x)
=
ke the one
zontal ta
Ordinate of
esponding
point with
alue attainet
factor
like a horizontal line (as in Fig. 3.5.1
in this manner?
[2,3]
(0,1)
2 f(x)=2x+1: [-1.1)
1
4. f(x)=√ (0.1)
6. f(x)=5-²: 1-1,2)
*(1-x)
In Problems 11 through 40, find the maximum and minimum
values attained by the given function on the indicated closed
interval
11. f(x) = 3x-2;
[-2.3]
12 f(x)=4-3x: [-1.5]
13. h(x)=4-x²¹: [1.3]
14. f(x)=x²+3; [0.5]
28. f(x)-(2x-3): (1.2)
29. f(x)-5+17-3x: (1.5)
30. f(x)=x+11+x-11: (-2,23
31. f(x)=50-105x²+72x: 10.13
32. fux)=2x+ (1.4)
33700)= 10.31
34. f(x)=110,3)
35. f(x)=(-2,5)
*+1
56. fox)-2- (-1.5
37. f(x)=x√1-¹ (-11)
38. f(x)=x√/4-¹ (0.2)
39. f(x)=x2-x)
40. f(x)=x2-x¹²,
1 g(x)=(x-1): [-1.4]
16. h(x)=x² + 4x +7: [-3.0]
17. f(x)=x²-3x: [-2,4]
18 g(x)=2r³-9x² + 12x: (0.4]
19. h(x)=x+ [1.4]
16
[1.3]
21. f(x)=3-2x; [-1.1]
23. f(x)=5-12x-9x²; [-1,1]
22. f(x)=x²-4x+3; [0,2]
24. f(x)=2x² - 4x +7; [0.2]
25. f(x)=x²-3x²-9x+5; [-2.4]
ed interval (a. 20. f(x)=x²+ =
(if any) of f.
minimum nor a ka
ossibility that fs
ferentiable there
xtremum in (a, b)
m, but only one cri
local minimum, and
in (a, b).
26. f(x)=³+x: [-1.2]
27. f(x)=3x³-5x³; [-2.2]
(1.3)
(0.4)
41. Suppose that fix) Ax+B is a linear function and that
A 0. Explain why the maximum and minimum values of
fon a closed interval (a, b) must occur at the endpoints of
the interval.
42. Suppose that is continuous on [a, b] and differentiable
on (a, b) and that f'(x) is never zero at any point of (a, b).
Explain why the maximum and minimum values of f must
occur at the endpoints of the interval (a, b).
43. Explain why every real number is a critical point of the
greatest integer function f(x)=[x].
44. Prove that every quadratic function
f(x) = ax²+bx+c (20)
has exactly one critical point on the real line.
45. Explain why the cubic polynomial function
f(x) = ax + bx² +cx+d (a*0)
can have either two, one, or no critical points on the real line.
Produce examples that illustrate each of the three cases.
46. Define f(x) to be the distance from x to the nearest integ
What are the critical points of f
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