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📝 The Extreme Value Theorem

📖 From Calculus • 5. The derivative in graphing and applications • 26 questions available

Practice MCQs for The Extreme Value Theorem. Test your knowledge with carefully crafted questions across easy, medium, and hard difficulty levels.

🔄 Last updated: 2026-07-15

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📝 Sample Questions

Q1. Consider \(f(x)=\sqrt{x}\) on the interval \([0,4]\). Which statement correctly applies the EVT?

🔹 A. \(f\) attains its maximum at \(x=0\).

🔹 B. \(f\) attains its minimum at \(x=4\).

🔹 C. \(f\) attains its maximum at \(x=4\) and its minimum at \(x=0\).

🔹 D. \(f\) does not satisfy the EVT because it is not differentiable at \(x=0\).

💡 Difficulty: easy | ✅ Correct: C

Q2. If a function is continuous on \([a,b]\) but not differentiable at a point \(c\in(a,b)\), what does the EVT say about the existence of extrema at \(c\)?

🔹 A. \(c\) cannot be an extremum because differentiability is required.

🔹 B. \(c\) may be an absolute extremum even if the derivative does not exist there.

🔹 C. The EVT does not apply because differentiability fails.

🔹 D. Only local extrema can occur at \(c\).

💡 Difficulty: medium | ✅ Correct: B

Q3. In a proof of the EVT using the completeness of \(\mathbb{R}\), which theorem is typically invoked to locate a point where the supremum is attained?

🔹 A. Bolzano–Weierstrass Theorem

🔹 B. Mean Value Theorem

🔹 C. Intermediate Value Theorem

🔹 D. Fundamental Theorem of Calculus

💡 Difficulty: hard | ✅ Correct: A

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🔗 Related Topics

📝 Approximating Roots 📝 Areas and Limits 📝 Continuity in Applications 📝 Continuity of Compositions 📝 Continuity of Inverse Functions

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