Theorem Let D be a closed and bounded subset of R. If f: DR is a continuous function, then the set f(D) is closed and bounded. Theorem Let S be a subset of real numbers. If S is closed and bounded, then it has a maximum and a minimum. Equivalently, if S is sequentially compact, then it has a maximum and a minimum. Theorem Extreme Value Theorem Let D be a closed and bounded subset of R. If f DR is a continuous function, then f has a maximum value and a minimum value.
Theorem Let D be a closed and bounded subset of R. If f: DR is a continuous function, then the set f(D) is closed and bounded. Theorem Let S be a subset of real numbers. If S is closed and bounded, then it has a maximum and a minimum. Equivalently, if S is sequentially compact, then it has a maximum and a minimum. Theorem Extreme Value Theorem Let D be a closed and bounded subset of R. If f DR is a continuous function, then f has a maximum value and a minimum value.
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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prove extreme value theorem with given information
Transcribed Image Text:Theorem
Let D be a closed and bounded subset of R. If f: D→ R is a continuous
function, then the set f(D) is closed and bounded.
Theorem
Let S be a subset of real numbers. If S is closed and bounded, then it has a
maximum and a minimum. Equivalently, if S is sequentially compact, then
it has a maximum and a minimum.
Theorem
Extreme Value Theorem
Let D be a closed and bounded subset of R. If f: D→ R is a continuous
function, then f has a maximum value and a minimum value.
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