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Rolle's theorem - Quiz
  • 1. If a function is convex or concave, what exists at every inner point?
A) Right- and left-hand derivatives
B) Only the left-hand derivative
C) Only the right-hand derivative
D) No derivatives exist
  • 2. When f(a) = f(b), where must f attain its maximum or minimum?
A) Somewhere between a and b
B) Outside the interval [a, b]
C) At exactly a or b
D) Only at the endpoints
  • 3. What happens if both the maximum and minimum are attained at the endpoints of [a, b]?
A) The derivative is undefined
B) The function is strictly increasing
C) The function is strictly decreasing
D) The function is constant on [a, b]
  • 4. In the proof, what does f(c + h) - f(c) / h evaluate to when h > 0 for a maximum at c?
A) Indeterminate
B) Greater than or equal to 0
C) Less than or equal to 0
D) Exactly 1
  • 5. What is the result of the left-hand derivative f'(c-) if c is a maximum point?
A) Always zero
B) Negative infinity
C) Less than or equal to 0
D) Greater than or equal to 0
  • 6. If the right- and left-hand limits agree, what is true about the derivative at c?
A) It is negative
B) It must be zero
C) It is undefined
D) It is positive
  • 7. What theorem can be applied as an alternative to the proof provided?
A) The mean value theorem
B) Bolzano's theorem
C) Taylor's theorem
D) Fermat's stationary point theorem
  • 8. For x < y, which condition proves convexity regarding one-sided derivatives?
A) f'(x+) = f'(y-)
B) f'(x-) <= f'(x+) <= f'(y-)
C) f'(x-) > f'(y-)
D) f'(x+) >= f'(x-) >= f'(y+)
  • 9. What kind of function must f be on the closed interval [a, b] according to the generalization of Rolle's theorem?
A) Non-real
B) Step-wise
C) Continuous
D) Discontinuous
  • 10. In the generalization, the limits of the derivative exist in which set?
A) The set of all rational numbers
B) The extended real line [-∞, ∞]
C) The set of all complex numbers
D) Only the interval (0, 1)
  • 11. The left-hand limit is defined as the limit as h approaches what?
A) 0+
B) a
C) 0-
D) b
  • 12. Who is the theorem named after?
A) Cauchy
B) Moritz Wilhelm Drobisch
C) Bhāskara II
D) Michel Rolle
  • 13. In what century did Bhāskara II state an early form of Rolle's theorem?
A) 18th century
B) 12th century
C) 19th century
D) 17th century
  • 14. What did Michel Rolle's 1691 proof specifically cover?
A) Exponential functions
B) Absolute value functions
C) Polynomial functions
D) Trigonometric functions
  • 15. Who first proved the theorem in 1823 as a corollary of the mean value theorem?
A) Bhāskara II
B) Michel Rolle
C) Cauchy
D) Giusto Bellavitis
  • 16. When was the name 'Rolle's theorem' first used by Moritz Wilhelm Drobisch?
A) 1823
B) 1834
C) 1691
D) 1846
  • 17. Is differentiability at the endpoints required for Rolle's theorem?
A) Yes, but only for polynomials
B) Yes, always
C) Only if the function is a semicircle
D) No
  • 18. Which mathematician first used the name 'Rolle's theorem' in 1846?
A) Bhāskara II
B) Michel Rolle
C) Cauchy
D) Giusto Bellavitis
  • 19. Why did Michel Rolle avoid using methods of differential calculus in his 1691 proof?
A) He preferred geometry
B) They had not been invented yet
C) He lacked formal training in calculus
D) He considered them fallacious
  • 20. What is the domain of the example function f defined in the text?
A) Rational numbers
B) Complex numbers
C) Real numbers
D) Integers
  • 21. What mathematical technique is used to prove the generalization of Rolle's theorem for higher derivatives?
A) Integration by parts
B) Mean value theorem
C) Contradiction
D) Mathematical induction
  • 22. For a function to satisfy the generalization of Rolle's theorem for the nth derivative, how many intervals with equal function values are required?
A) 2 intervals
B) n intervals
C) n+1 intervals
D) n-1 intervals
  • 23. What is the standard name for the property where a field's polynomials have roots such that their derivatives also have roots in the same field?
A) Voorhoeve property
B) Kaplansky's property
C) Differentiable property
D) Rolle's property
  • 24. What is the base case for the induction proof of the generalized Rolle's theorem?
A) n = 2
B) n = 1
C) n = 0
D) The limit as n approaches infinity
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