Exercise · ZW-EXTREME-VALUE-THEOREM-020 · ★★★ Advanced
Let \(f\) be continuous on \([a,b]\) and \(M=\sup\{f(x):x\in[a,b]\}\). What is true?
Choose one option.
Show answer
- D. There is \(c\in[a,b]\) with \(f(c)=M\)
Explanation
The theorem ensures the supremum is a maximum: some point \(c\) attains it. Failing to attain it only happens when continuity or closedness is missing.
Review the concept: Extreme value theorem
If \(f\) is continuous on a closed and bounded interval \([a,b]\), then \(f\) attains an absolute maximum and minimum on \([a,b]\).
\[f\ \text{continuous on }[a,b]\ \Rightarrow\ \exists\,m,M\in[a,b]:\ f(m)\le f(x)\le f(M)\]
More practice problems: extreme value theorem
- ZW-EXTREME-VALUE-THEOREM-001 For which of these functions is an absolute maximum NOT guaranteed by the extreme value theorem?
- ZW-EXTREME-VALUE-THEOREM-002 Which condition of the extreme value theorem fails for f(x)=(1)/(x) on [-1,1]?
- ZW-EXTREME-VALUE-THEOREM-003 What does the extreme value theorem guarantee for a function continuous on [a,b]?
- ZW-EXTREME-VALUE-THEOREM-004 Which of these intervals is closed and bounded?
- ZW-EXTREME-VALUE-THEOREM-005 What does it mean for f to have an absolute maximum on [a,b]?
- ZW-EXTREME-VALUE-THEOREM-006 Which of these functions is NOT continuous on [0,1]?
- ZW-EXTREME-VALUE-THEOREM-007 Let f(x)=x^3-3x on [-2,2]. What is the absolute maximum value of f on that interval?
- ZW-EXTREME-VALUE-THEOREM-008 Which of these functions shows that the theorem fails when the function is not continuous?