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Prove (c) Consider the function ƒ : B₁(0) → R given by the formula ƒ(T₁, T2, T3) = £₁ + e¹--1--3--} . that f attains its minimum on B₁ (0) but does not attain its supremum on B₁ (0).

Prove (c) Consider the function ƒ : B₁(0) → R given by the formula ƒ(T₁, T2, T3) = £₁ + e¹--1--3--} . that f attains its minimum on B₁ (0) but does not attain its supremum on B₁ (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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i want you to solve part C. remember from b, use the conclusion there, try to show as x in this open ball apporaching unit sphere, f should go to infinity.  you prove that all boundary points goes to infinity. 

4. (a) Let A CR" be a set such that A ‡ Ø. Let ε : A → (0, ∞) be a function on A taking values in the positive reals. Prove that A \ (U₁€@A Bɛ(a)(a)) is closed. (b) Let A CR" be non-empty, open, and bounded. Let ƒ : A → R be a continuous function such that lim f(x) = ∞ for all a € A. Prove that f attains its minimum on A. x→a (e) Consider the function ƒ : B₁(0) → R given by the formula ƒ(x₁, 72, 73) = £₁ + e¹---¹--³. Prove that f attains its minimum on B₁ (0) but does not attain its supremum on B₁(0).
Transcribed Image Text:4. (a) Let A CR" be a set such that A ‡ Ø. Let ε : A → (0, ∞) be a function on A taking values in the positive reals. Prove that A \ (U₁€@A Bɛ(a)(a)) is closed. (b) Let A CR" be non-empty, open, and bounded. Let ƒ : A → R be a continuous function such that lim f(x) = ∞ for all a € A. Prove that f attains its minimum on A. x→a (e) Consider the function ƒ : B₁(0) → R given by the formula ƒ(x₁, 72, 73) = £₁ + e¹---¹--³. Prove that f attains its minimum on B₁ (0) but does not attain its supremum on B₁(0).
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