Exercise · ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-006 · ★★☆ Intermediate
What is the maximum of \(f(x)=x(3-x)\) on \([0,3]\)?
Choose one option.
Show answer
- C. \(9/4\)
Explanation
Since \([0,3]\) is compact, \(f\) attains its maximum. \(f'(x)=3-2x=0\) gives \(x=3/2\) and \(f(3/2)=9/4\), larger than \(f(0)=f(3)=0\).
Review the concept: Continuous image of a compact space
If \(f:X\to Y\) is continuous and the space \(X\) is compact, then the image \(f(X)\) is a compact subspace of \(Y\).
\[f(X)\ \text{compact}\]
More practice problems: continuous image of a compact space
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- ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-007 What is the minimum of f(x)=x^3-3x on [0,3]?
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