Exercise · ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-007 · ★★☆ Intermediate
What is the minimum of \(f(x)=x^3-3x\) on \([0,3]\)?
Choose one option.
Show answer
- A. \(-2\)
Explanation
The minimum exists by compactness and lies at an endpoint or a critical point. \(f'(x)=3x^2-3=0\) gives \(x=1\), with \(f(1)=-2\); also \(f(0)=0\) and \(f(3)=18\).
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
- ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-001 Let f:[0,1]→ℝ, f(x)=x^2+1. What is f([0,1])?
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- ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-006 What is the maximum of f(x)=x(3-x) on [0,3]?
- ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-008 Let f:X→ Y be continuous and surjective with X compact. What follows?
- ZW-CONTINUOUS-IMAGE-OF-A-COMPACT-SPACE-009 Let X be compact and f:X→ℝ continuous with f(x)>0 for all x. Is there c>0 with f(x)≥ c on all of X?
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