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…

Assume that f: [0, ∞) → R is a continuous function such that lim ƒ(x) = ∞. x48 a) Prove that f admits a minimum on [0, ∞). o) Does the statement of part (a) remain true in general if we remove the condition lim f(x) = x48 co? Justify your answer.

Assume that f: [0, ∞) → R is a continuous function such that lim ƒ(x) = ∞. x48 a) Prove that f admits a minimum on [0, ∞). o) Does the statement of part (a) remain true in general if we remove the condition lim f(x) = x48 co? Justify your answer.

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.3: More On Functions; Piecewise-defined Functions

Problem 99E: Determine if the statemment is true or false. If the statement is false, then correct it and make it...

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Question

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.

Branch of mathematical analysis that studies real numbers, sequences, and series of real numbers and real functions. The concepts of real analysis underpin calculus and its application to it. It also includes limits, convergence, continuity, and measure theory.

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 $limit as x rightwards arrow infinity of f open parentheses x close parentheses equals infinity$ .

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