Sketch the graph of f with emphasis on concavity. Label asymptotes with their equations and important points with coordinates.
Sketch the graph of f with emphasis on concavity. Label asymptotes with their equations and important points with coordinates.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Sketch the graph of f with emphasis on concavity. Label asymptotes with their equations and important points with coordinates.
Transcribed Image Text:f'
f"
conclusions
(-∞, -1) -1 (-1,1)
+
+
+
0
+
1 (1, 2) 2
-
0
dne
1
dne
(2,3)
+
3 (3, +00)
0
+
t
+
![e
is continuous everywhere except at x = 2
f(-1)=-2, f(1) = 0,
f(3) = 5
lim
f(x) = -1, lim
f(x) = +1
2-2-
11-2+
lim [f(x) = (x+1)] = 0
1
2
(x,-1).
f'
+.
O
Idne
f!!
+
O
Idne
+
(1) then fis increasing in (-4, 1)
becaum, fix) > 0 in (-4,1)
increasing
Also, f'(x) >0 in (3,4) =) f is also in (3, d) U(-1,1)
^
fis increasing in (-α₁1) U (3,α) u (-1,1)
(2)
@f!(~) 2.0 in (-1,2) U(213)
a fis decreasing
in (1,2) 0 (2,3)
F"(x) >0 for all x = (- ~, -1) U (2/0)
F
f(x) 20 for all 2€ (-1, 2)
⇒ fis
(3)
ه و
(4)
+
(-1, 1)
lim fex) = -3,
2-22
3 را
2 را
-
3 3,4
O
+
+
f is concave down for all x = (-4,-1) U (2,4)
concave down forall: x=(-1,2)
(St) Also,
x =
(6) point of inflection are x= -1; 2;
+
are the point of extreama.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F67d9a3aa-f23c-475e-b66e-be3e4e699f57%2Fd074b559-4722-40c5-a1c4-b5f9e6b8fd67%2Fh4exyit_processed.jpeg&w=3840&q=75)
Transcribed Image Text:e
is continuous everywhere except at x = 2
f(-1)=-2, f(1) = 0,
f(3) = 5
lim
f(x) = -1, lim
f(x) = +1
2-2-
11-2+
lim [f(x) = (x+1)] = 0
1
2
(x,-1).
f'
+.
O
Idne
f!!
+
O
Idne
+
(1) then fis increasing in (-4, 1)
becaum, fix) > 0 in (-4,1)
increasing
Also, f'(x) >0 in (3,4) =) f is also in (3, d) U(-1,1)
^
fis increasing in (-α₁1) U (3,α) u (-1,1)
(2)
@f!(~) 2.0 in (-1,2) U(213)
a fis decreasing
in (1,2) 0 (2,3)
F"(x) >0 for all x = (- ~, -1) U (2/0)
F
f(x) 20 for all 2€ (-1, 2)
⇒ fis
(3)
ه و
(4)
+
(-1, 1)
lim fex) = -3,
2-22
3 را
2 را
-
3 3,4
O
+
+
f is concave down for all x = (-4,-1) U (2,4)
concave down forall: x=(-1,2)
(St) Also,
x =
(6) point of inflection are x= -1; 2;
+
are the point of extreama.
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