Real Analysis, Continuous functions Exercise 0.3.Assume that f : [0, 00) → R is a continuous function such that lim f(x) == 0.(a) Prove that f admits a minimum on [0, 0).(b) Does the statement of part (a) remain true in general if we remove the condition lim f(x) =0? Justify your answer.

Given that is a continuous function such that .(a)We need to prove that attain its minimum value…

Answered: Assume that f: [0, ∞) → R is a…

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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Real Analysis, Continuous functions

Exercise 0.3.
Assume that f : [0, 00) → R is a continuous function such that lim f(x) =
= 0.
(a) Prove that f admits a minimum on [0, 0).
(b) Does the statement of part (a) remain true in general if we remove the condition lim f(x) =
0? Justify your answer.

Expert Solution

Step 1: Introduction

Given that $f colon stretchy left square bracket 0 comma infinity stretchy right parenthesis rightwards arrow straight real numbers$ is a continuous function such that $limit as x rightwards arrow infinity of f open parentheses x close parentheses equals infinity$ .

(a)

We need to prove that $f$ attain its minimum value on $stretchy left square bracket 0 comma infinity stretchy right parenthesis$ .

(b)

We need to determine whether the statement (a) is still be true, if we remove the condition $limit as x rightwards arrow infinity of f open parentheses x close parentheses equals infinity$ .