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Exercise 2 (*). Recall that a function f (a,b) → R is increasing on (a, b) if f(x) ≤ f(y) whenever x 0. 2. Now, prove that g is not increasing over any open interval containing 0.

Exercise 2 (*). Recall that a function f (a,b) → R is increasing on (a, b) if f(x) ≤ f(y) whenever x 0. 2. Now, prove that g is not increasing over any open interval containing 0.

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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Exercise 2 (*). Recall that a function f (a,b) → R is increasing on (a, b) if f(x) ≤ f(y) whenever x <y in (a, b). A familiar mantra from calculus is that a differentiable function is increasing if its derivative is positive, but the statement required some sharpening in order to be completely accurate. To that end, show that the function g(x) = {{ 2 0 + x² sin(-) n() if x 70 if x = 0 1. Is differentiable on R and satisfies g'(0) > 0. 2. Now, prove that g is not increasing over any open interval containing 0.
Transcribed Image Text:Exercise 2 (*). Recall that a function f (a,b) → R is increasing on (a, b) if f(x) ≤ f(y) whenever x <y in (a, b). A familiar mantra from calculus is that a differentiable function is increasing if its derivative is positive, but the statement required some sharpening in order to be completely accurate. To that end, show that the function g(x) = {{ 2 0 + x² sin(-) n() if x 70 if x = 0 1. Is differentiable on R and satisfies g'(0) > 0. 2. Now, prove that g is not increasing over any open interval containing 0.
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