(c) Let f: RR be a function which is differentiable and such that f': R → R is bounded. Prove that f is uniformly continuous. (d) Let f be a differentiable function which is increasing in the sense that r ≤ y implies f(x) ≤ f(y). Prove that f'(a) ≥ 0 for all a. (e) Let f (a,b)→ R be a function such that f'(x) = 0 for all r = (a, b). Prove that f is a constant function.
(c) Let f: RR be a function which is differentiable and such that f': R → R is bounded. Prove that f is uniformly continuous. (d) Let f be a differentiable function which is increasing in the sense that r ≤ y implies f(x) ≤ f(y). Prove that f'(a) ≥ 0 for all a. (e) Let f (a,b)→ R be a function such that f'(x) = 0 for all r = (a, b). Prove that f is a constant function.
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
parts c, d, e
![(a) Define f: R→ R by
1. DERIVATIVES
[x² sin (-¹)
3²
f(x) = {
x #0
x = 0.
Prove that f is differentiable at all x and find the value f'(0). Prove f' is not continuous.
(b) Prove that for all z € [0, 1] we have
2
x(1 − 2)" <n ²7 1²
–
(c) Let f: R→ R be a function which is differentiable and such that f': R → R is bounded.
Prove that f is uniformly continuous.
(d) Let f be a differentiable function which is increasing in the sense that r ≤ y implies
f(x) ≤ f(y). Prove that f'(a) ≥ 0 for all a.
(e) Let f: (a,b) → R be a function such that f'(x) = 0 for all x = (a, b). Prove that f is a
constant function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F04625ac1-43ff-4999-b93a-55388fc0c5e2%2Fed1463aa-ae8b-4212-852e-9335b1245be2%2F3by02on_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Define f: R→ R by
1. DERIVATIVES
[x² sin (-¹)
3²
f(x) = {
x #0
x = 0.
Prove that f is differentiable at all x and find the value f'(0). Prove f' is not continuous.
(b) Prove that for all z € [0, 1] we have
2
x(1 − 2)" <n ²7 1²
–
(c) Let f: R→ R be a function which is differentiable and such that f': R → R is bounded.
Prove that f is uniformly continuous.
(d) Let f be a differentiable function which is increasing in the sense that r ≤ y implies
f(x) ≤ f(y). Prove that f'(a) ≥ 0 for all a.
(e) Let f: (a,b) → R be a function such that f'(x) = 0 for all x = (a, b). Prove that f is a
constant function.
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