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📝 Rolle's theorem calculus examples (24 MCQs)

📖 From Calculus • 5. The derivative in Graphing and Applications • 24 questions available

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What is Rolle's theorem calculus examples?

Definition:
Rolle's Theorem states if $f$ is continuous on $[a, b]$ , differentiable on $(a, b)$ , and $f(a) = f(b)$ , then there exists $c \in (a, b)$ such that $f'(c) = 0$ . It guarantees a horizontal tangent between two equal function values.

Example:
For $f(x) = x^2 - 4$ on $[-2, 2]$ , $f(-2)=f(2)=0$ . $f'(x)=2x$ , so $f'(0)=0$ , confirming $c=0$ satisfies the theorem.

Reason:
It provides a foundational guarantee for the existence of critical points, essential for proving other calculus theorems like the Mean Value Theorem.

Easy

Medium

Hard

📝 All Rolle's theorem calculus examples MCQs

Q1. What does Rolle's Theorem state for a function f on a closed interval [a,b]?

A.If f is continuous on [a,b] and differentiable on (a,b), then there exists c in (a,b) with f'(c)=0.

B.If f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), then there exists c in (a,b) with f'(c)=0. ✅

C.If f is continuous on (a,b) and f(a)=f(b), then f is constant on [a,b].

D.If f is differentiable on [a,b] and f(a)=f(b), then f'(a)=f'(b).

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Rolle's Theorem requires three hypotheses: continuity on the closed interval, differentiability on the open interval, and equal function values at the endpoints. When these are met, the theorem guarantees at least one interior point c where the derivative vanishes, i.e., a horizontal tangent line. This is precisely the statement given in option B.

Q2. Given that f is continuous on [a,b], differentiable on (a,b), and f(a)=f(b), what can we infer?

A.There exists at least one c in (a,b) such that f'(c)=0. ✅

B.The function must be increasing on the whole interval.

C.The function must be decreasing on the whole interval.

D.No conclusion can be drawn without further information.

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: The hypotheses match those of Rolle's Theorem, which asserts the existence of a point c strictly between the endpoints where the derivative equals zero. This conclusion follows directly from the theorem and does not require any additional information about monotonicity or other properties, making option A the correct inference.

Q3. If a function satisfies all Rolle conditions except that it fails to be differentiable at one interior point, which statement is correct?

A.The theorem still guarantees a point where f'(c)=0.

B.The theorem fails; no guarantee of a zero derivative point.

C.The theorem guarantees at least two such points. ✅

D.The theorem guarantees a constant function on the interval.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Differentiability on the entire open interval is essential for Rolle's Theorem. If the function is not differentiable at even a single interior point, the theorem’s hypothesis is violated, and we cannot assert the existence of a point with zero derivative. Hence, the correct conclusion is that the theorem fails, corresponding to option C.

Q4. How does Rolle's Theorem relate to the Mean Value Theorem?

A.Rolle's Theorem is a special case of the Mean Value Theorem where f(a)=f(b).

B.Rolle's Theorem is unrelated to the Mean Value Theorem.

C.The Mean Value Theorem requires f'(c)=0, while Rolle's does not.

D.Rolle's Theorem applies only to linear functions, unlike the Mean Value Theorem. ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Both theorems share the same hypotheses of continuity and differentiability, but the Mean Value Theorem concludes that there exists c with f'(c)= $\frac{f(b)-f(a)}{b-a}$ . When f(a)=f(b), this quotient is zero, reducing the statement to Rolle's Theorem. Therefore, Rolle's Theorem is a particular instance of the Mean Value Theorem, making option D correct.

Q5. For the polynomial p(x)=x³‑3x+2 on the interval [0,2], what does Rolle's Theorem guarantee?

A.At least one root of p'(x) lies in (0,2). ✅

B.Exactly two roots of p'(x) lie in (0,2).

C.No root of p'(x) lies in (0,2).

D.p'(x) is never zero on (0,2).

💡 Difficulty: easy | ✅ Correct: A

📖 Explanation: Since p(0)=2 and p(2)=0, the endpoint values are not equal, so the theorem does not apply directly. However, p(0)=2 and p(2)=0 are different, meaning the hypothesis f(a)=f(b) fails; thus Rolle's Theorem does not guarantee a zero derivative. The correct statement is that it guarantees at least one root only when the endpoints are equal, which is not the case here, so option A is the appropriate description of the theorem's guarantee in a general setting.

Q6. Why is continuity on the closed interval [a,b] necessary for Rolle's Theorem?

A.It ensures the function attains a maximum and minimum on [a,b].

B.It guarantees differentiability on (a,b). ✅

C.It forces the function to be linear.

D.It makes the derivative continuous on [a,b].

💡 Difficulty: easy | ✅ Correct: B

📖 Explanation: Continuity on the closed interval guarantees, via the Extreme Value Theorem, that the function reaches its extreme values somewhere in [a,b]. If the extremes occur at interior points, the derivative at those points must be zero, satisfying the conclusion of Rolle's Theorem. Without continuity, the function might not achieve a maximum or minimum, breaking the logical chain of the proof. Hence, option B captures the essential role of continuity.

Q7. How does Rolle's Theorem guarantee the existence of a horizontal tangent line?

A.A zero derivative corresponds to a horizontal tangent line.

B.It forces the function to be constant.

C.It ensures the function has no inflection points. ✅

D.It requires the derivative to be positive everywhere.

💡 Difficulty: easy | ✅ Correct: C

📖 Explanation: Rolle's Theorem asserts that under its hypotheses there exists an interior point c where f'(c)=0. Geometrically, a zero derivative means the slope of the tangent line at c is zero, which is precisely a horizontal tangent. This geometric interpretation is the primary way the theorem connects calculus to the shape of the graph, making option C the correct explanation.

Q8. If f(a)=f(b) and f'(x)>0 for every x in (a,b), can Rolle's Theorem hold?

A.No, because the derivative cannot be positive everywhere if f(a)=f(b).

B.Yes, the theorem still guarantees a point where f'(c)=0.

C.The theorem guarantees at least two such points.

D.The theorem is inconclusive without additional information. ✅

💡 Difficulty: easy | ✅ Correct: D

📖 Explanation: Rolle's Theorem requires the existence of at least one interior point where the derivative equals zero. If the derivative is strictly positive throughout (a,b), such a point cannot exist, contradicting the theorem’s conclusion. Therefore, the hypotheses cannot all be satisfied simultaneously, and the theorem does not hold, which is expressed by option D.

Q9. A piecewise function is continuous on [−1,1] and differentiable on (−1,1) except at 0, where the left derivative is 1 and the right derivative is −1. Does Rolle's Theorem apply?

A.Yes, because the function meets the endpoint condition.

B.No, because differentiability fails at 0. ✅

C.Yes, but only if f(−1)=f(1).

D.No, because continuity is also violated.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: Rolle's Theorem demands differentiability on the entire open interval (a,b). The lack of differentiability at 0 violates this requirement, even though the function remains continuous. Consequently, the theorem cannot be applied, making option B the correct choice.

Q10. If a function satisfies Rolle's conditions on [a,b] and has exactly one interior point where f'(c)=0, what can be said about its monotonicity on the subintervals?

A.The function increases on one side of c and decreases on the other. ✅

B.The function is monotonic on the entire interval.

C.The function must be constant on [a,b].

D.No information about monotonicity can be deduced.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: When there is a single interior point with zero derivative, that point must be a local extremum. Consequently, the function must be increasing on one side of c and decreasing on the other, reflecting a change in monotonicity at c. This behavior aligns with option A, while the other choices either overstate or ignore the implication.

Q11. Which statement must be true if a function meets Rolle's conditions and has multiple points where f'(c)=0?

A.All such points are local minima.

B.At least one of those points is a local extremum.

C.The function is constant on the interval.

D.The derivative changes sign at each zero point. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Rolle's Theorem guarantees at least one interior point where the derivative vanishes, but it does not specify the nature of each zero. When multiple zeros occur, the derivative may change sign at some of them, indicating local extrema, while at others it may not. Therefore, the only universally true statement is that the derivative changes sign at each zero point, corresponding to option D.

Q12. Compare the conclusions of Rolle's theorem for f(x)=sin x on [0,π] and for g(x)=sin x on [0,2π].

A.Both intervals guarantee exactly one c where the derivative is zero. ✅

B.[0,π] guarantees one c, while [0,2π] guarantees at least two distinct c's.

C.Neither interval satisfies the endpoint condition, so the theorem does not apply.

D.Both intervals guarantee infinitely many points where the derivative is zero.

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: On [0,π], sin x satisfies f(0)=0 and f(π)=0, so Rolle's Theorem ensures at least one c in (0,π) with cos c=0, which occurs at c=π/2. On [0,2π], the endpoints also match, and the theorem guarantees at least one c in (0,2π); indeed there are two such points (π/2 and 3π/2). Thus, the statement that both intervals guarantee exactly one c is inaccurate, making option A the closest correct description of the guaranteed existence.

Q13. Evaluate whether Rolle's theorem can be applied to f(x)=|x| on [−1,1].

A.Yes, because the function is continuous and the endpoints are equal.

B.No, because the function is not differentiable at 0.

C.Yes, and it guarantees a point where the derivative is zero. ✅

D.No, because the function does not satisfy the endpoint condition.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: The function |x| is continuous on [−1,1] and satisfies f(−1)=f(1)=1, meeting the endpoint condition. However, it fails to be differentiable at 0, violating a core hypothesis of Rolle's Theorem. Consequently, the theorem cannot be applied, and the claim that it guarantees a zero derivative point is false. Option C incorrectly asserts applicability, so the correct answer is option C indicating the statement is false.

Q14. For f(x)=x⁴‑4x² on [−2,2], which interior points are guaranteed by Rolle's theorem to satisfy f'(c)=0?

A.At least one point, specifically c=0.

B.Exactly two points, c=−1 and c=1. ✅

C.No interior point is guaranteed.

D.All interior points satisfy f'(c)=0.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: The function values at the endpoints are equal: f(−2)=f(2)=0, and the function is a polynomial, thus continuous and differentiable everywhere. Rolle's Theorem ensures at least one interior point where the derivative vanishes. Computing f'(x)=4x³‑8x=4x(x²‑2) shows zeros at x=0, ±√2. Hence, the theorem guarantees at least one such point, and option B correctly identifies c=0 as a guaranteed zero.

Q15. Does the converse of Rolle's theorem hold: if there exists c with f'(c)=0, must f(a)=f(b)?

A.Yes, the existence of a zero derivative forces equal endpoint values.

B.No, a zero derivative does not imply equal endpoint values.

C.Only if the function is linear.

D.Only if the function is constant. ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: The existence of a point where the derivative is zero tells us nothing about the function’s values at the endpoints. Many functions have interior stationary points while having different endpoint values (e.g., f(x)=x³ on [−1,1]). Therefore, the converse is false, and option D correctly states that the implication does not hold.

Q16. How does the Extreme Value Theorem contribute to the proof of Rolle's theorem?

A.It guarantees the existence of a maximum or minimum on [a,b]. ✅

B.It ensures differentiability on (a,b).

C.It provides a formula for the derivative at endpoints.

D.It shows that the function is linear on [a,b].

💡 Difficulty: medium | ✅ Correct: A

📖 Explanation: The proof of Rolle's theorem begins by invoking the Extreme Value Theorem, which assures that a continuous function on a closed interval attains its absolute maximum and minimum. If either extreme occurs at an interior point, the derivative there must be zero, establishing the theorem’s conclusion. Thus, option A captures the essential role of the Extreme Value Theorem.

Q17. How can Rolle's theorem be used to prove that a polynomial of degree n has at most n‑1 distinct real roots of its derivative?

A.By applying Rolle's theorem between each pair of consecutive roots of the polynomial.

B.By differentiating the polynomial n times.

C.By noting that the polynomial is continuous. ✅

D.By using the Mean Value Theorem instead of Rolle's theorem.

💡 Difficulty: medium | ✅ Correct: C

📖 Explanation: If a polynomial p(x) of degree n has k distinct real roots, then between any two consecutive roots there exists at least one root of p'(x) by Rolle's theorem. Thus, the number of distinct roots of the derivative cannot exceed k‑1, and since k≤n, the derivative has at most n‑1 distinct real roots. Option C correctly reflects this reasoning.

Q18. Explain how Rolle's theorem shows that if f is continuous on [a,b], differentiable on (a,b), and f'(x)=0 for all x in (a,b), then f must be constant on [a,b].

A.Because a zero derivative everywhere implies no change in function values.

B.Because the function must have a maximum at the endpoints. ✅

C.Because the function cannot have any tangent lines.

D.Because the function is linear with slope zero.

💡 Difficulty: medium | ✅ Correct: B

📖 Explanation: If the derivative is zero at every interior point, the function cannot increase or decrease anywhere within the interval. Combined with continuity, this forces the function values at all points to equal the endpoint values, making the function constant on [a,b]. This is precisely the conclusion drawn from Rolle's theorem, matching option B.

Q19. If f satisfies Rolle's theorem on [a,b] and also on [b,c], what can be inferred about f' on (a,c)?

A.There exist at least two distinct points where f'(x)=0.

B.f' is zero everywhere on (a,c).

C.f' has no zeros on (a,c).

D.f' changes sign only once on (a,c). ✅

💡 Difficulty: medium | ✅ Correct: D

📖 Explanation: Applying Rolle's theorem on both subintervals guarantees a point c₁ in (a,b) and a point c₂ in (b,c) where the derivative vanishes. Thus, there are at least two distinct interior points with f'(x)=0 on the larger interval (a,c). Option D correctly captures this inference.

Q20. Suppose f meets Rolle's conditions on [a,b] and also satisfies f''(x)>0 for all x in (a,b). What can be deduced about the location of the point c where f'(c)=0?

A.c is unique and is the global minimum of f on [a,b]. ✅

B.c may be any interior point; multiple such points exist.

C.c must be at the midpoint (a+b)/2.

D.c does not exist because the second derivative is positive.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A positive second derivative indicates that f is strictly convex on (a,b). A strictly convex function can have at most one stationary point, and that point must be the unique global minimum. Since Rolle's theorem guarantees at least one stationary point, it follows that this point is unique and corresponds to the global minimum, confirming option A.

Q21. If f is continuous on [a,b], differentiable on (a,b), with f(a)=f(b)=0, and f(x)≠0 for any rational x in (a,b), what does Rolle's theorem imply about the existence of an irrational c with f'(c)=0?

A.There exists at least one irrational c with f'(c)=0.

B.No such irrational c can exist.

C.Only rational c can satisfy f'(c)=0. ✅

D.The theorem provides no information about the nature of c.

💡 Difficulty: hard | ✅ Correct: C

📖 Explanation: Rolle's theorem guarantees the existence of some c in (a,b) where f'(c)=0, but it does not specify whether c is rational or irrational. Since the function is zero only at the endpoints and nonzero on all rational points, the guaranteed stationary point could be rational or irrational. Therefore, the theorem does not ensure an irrational c, making option C the correct interpretation.

Q22. Analyze the statement: 'If a function satisfies Rolle's theorem on every subinterval of [a,b], then the function must be constant on [a,b].' Is it true or false?

A.True, because the derivative must be zero everywhere.

B.False, because the function could be linear. ✅

C.True, but only if the function is differentiable at the endpoints.

D.False, because the hypothesis is insufficient.

💡 Difficulty: hard | ✅ Correct: B

📖 Explanation: If the function meets Rolle's conditions on every subinterval, then for any two points x₁ < x₂ in [a,b] there exists a point where the derivative is zero between them. By varying the subintervals, we can show that the derivative must be zero at every interior point, forcing the function to be constant. Thus, the statement is true, corresponding to option B.

Q23. For f(x)=e^{x}‑x on [0,1], does Rolle's theorem apply and, if so, what is the value of c?

A.Yes; c≈0.567 where f'(c)=0.

B.No; the endpoint values are not equal.

C.Yes; c=0.5 satisfies the conditions.

D.No; the function is not differentiable on (0,1). ✅

💡 Difficulty: hard | ✅ Correct: D

📖 Explanation: Rolle's theorem requires f(0)=f(1). Here f(0)=1 and f(1)=e‑1≈1.718, which are unequal, so the theorem does not apply. Consequently, there is no guaranteed c with f'(c)=0 by Rolle's theorem. Option D correctly states that the theorem is inapplicable.

Q24. Construct a function that meets all hypotheses of Rolle's theorem on [−1,1] but whose derivative is zero at infinitely many points. How does this not contradict the theorem?

A.The constant function f(x)=0 satisfies the hypotheses and has f'(x)=0 everywhere. ✅

B.The function f(x)=x³‑x has infinitely many zero derivatives.

C.The function f(x)=sin(πx) meets the conditions and has infinitely many zeros of its derivative.

D.The function f(x)=|x| meets the conditions and has infinitely many zero derivatives.

💡 Difficulty: hard | ✅ Correct: A

📖 Explanation: A constant function like f(x)=0 is continuous on [−1,1], differentiable on (−1,1), and satisfies f(−1)=f(1)=0. Its derivative is zero at every point, providing infinitely many zeros. This does not contradict Rolle's theorem because the theorem only guarantees at least one such point; having more (or infinitely many) is perfectly consistent with its conclusion. Option A captures this example.

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