Problem 129. Show that if f'(x) <0 for every x in the interval (a, b) then f is decreasing on (a, b).
Problem 129. Show that if f'(x) <0 for every x in the interval (a, b) then f is decreasing on (a, b).
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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Problem #129
![Theoren!"
such
Theorem 22. (The Mean Value Theorem) Suppose f' crists for crery
E (a.b) andf is contimuous on a, b). Then there is a real number CE (a, b)
of d
avege rate
f(b) – f(a) = ÖVer Mtenad q.b.
rove
ち-a f'lc) =
%3D
b- a
Theorem 23. (Rolle's Theorem) Suppose f' crists for crery r E (a.b). f
is continuous on a. b), and
Use to
prove
muT
to
f(a) = f(b).
-ve
Then there is a real number c E (a, b) such that
f'(c) = 0.
Problem 129. Show that if f'(x) <0 for every x in the interval (a, b) then f
is decreasing on (a, b).
Nested Interval Property of the Real Number System (NIP). Suppose
we have two sequences of real numbers (xn) and (yn) satisfying the following
conditions:
Chapte
1. 21 < a2 < a3 s... [(*n) is non-decreasing]
2. y1 2 y2 2 y3 2... [(yn) is non-increasing]
3. V n, In SYn
4. limn00 (Yn – an) = 0](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2F9e3bbdb2-70d0-42e6-bb6c-6eedc6571a9f%2F5y792oc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Theoren!"
such
Theorem 22. (The Mean Value Theorem) Suppose f' crists for crery
E (a.b) andf is contimuous on a, b). Then there is a real number CE (a, b)
of d
avege rate
f(b) – f(a) = ÖVer Mtenad q.b.
rove
ち-a f'lc) =
%3D
b- a
Theorem 23. (Rolle's Theorem) Suppose f' crists for crery r E (a.b). f
is continuous on a. b), and
Use to
prove
muT
to
f(a) = f(b).
-ve
Then there is a real number c E (a, b) such that
f'(c) = 0.
Problem 129. Show that if f'(x) <0 for every x in the interval (a, b) then f
is decreasing on (a, b).
Nested Interval Property of the Real Number System (NIP). Suppose
we have two sequences of real numbers (xn) and (yn) satisfying the following
conditions:
Chapte
1. 21 < a2 < a3 s... [(*n) is non-decreasing]
2. y1 2 y2 2 y3 2... [(yn) is non-increasing]
3. V n, In SYn
4. limn00 (Yn – an) = 0
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