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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
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
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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