When Rolle’s Theorem is verified for f(x) on [a, b] then there exists c such thata)c ε [a, b] such that f'(c) = 0b)c ε (a, b) such that f'(c) = 0c)c ε (a, b] such that f'(c) = 0d)c ε [a, b) such that f'(c) = 0Correct answer is option 'B'. Can you explain this answer?

Question

When Rolle&#8217;s Theorem is verified for f(x) on [a, b] then there exists c such that&#8203;a)c &#949; [a, b] such that f'(c) = 0b)c &#949; (a, b) such that f'(c) = 0c)c &#949; (a, b] such that f'(c) = 0d)c &#949; [a, b) such that f'(c) = 0Correct answer is option 'B'. Can you explain this answer?

Answer

Answer is
B) c ∈ (a, b) such that f'(c) = 0.
Statement for Rolle’s Theorem :
Suppose that a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Then if f(a)=f(b), then there exists at least one point c in the open interval (a,b) for which f′(c)=0.

Answer

Answer is
B) c ∈ (a, b) such that f'(c) = 0.
Statement for Rolle’s Theorem :
Suppose that a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Then if f(a)=f(b), then there exists at least one point c in the open interval (a,b) for which f′(c)=0.

Answer

Answer is
B) c ∈ (a, b) such that f'(c) = 0.
Statement for Rolle’s Theorem :
Suppose that a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Then if f(a)=f(b), then there exists at least one point c in the open interval (a,b) for which f′(c)=0.

Answer

Explanation:
To understand why option 'B' is the correct answer, let's first understand what Rolle's Theorem states.

Rolle's Theorem:
If a function f(x) is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), and if f(a) = f(b), then there exists at least one number c in the open interval (a, b) such that f'(c) = 0.

Key Points:
- Rolle's Theorem is a special case of the Mean Value Theorem.
- The conditions for applying Rolle's Theorem are:
- f(x) is continuous on the closed interval [a, b].
- f(x) is differentiable on the open interval (a, b).
- f(a) = f(b).

Explanation of Options:
a) There exists c in [a, b] such that f(c) = 0: This option is not true because Rolle's Theorem does not guarantee that f(c) equals zero, it only guarantees that f'(c) equals zero.

b) There exists c in (a, b) such that f(c) = 0: This option is the correct answer. Even though Rolle's Theorem does not directly state that f(c) equals zero, it implies that there exists at least one point c in the open interval (a, b) where the derivative of the function is zero. This means that the function may have a root or a zero at that point.

c) There exists c in (a, b] such that f(c) = 0: This option is not true because Rolle's Theorem guarantees the existence of a point in the open interval (a, b) where the derivative of the function is zero, not necessarily at the right endpoint b.

d) There exists c in [a, b) such that f(c) = 0: This option is not true because Rolle's Theorem guarantees the existence of a point in the open interval (a, b) where the derivative of the function is zero, not necessarily at the left endpoint a.

Conclusion:
The correct answer is option 'B' because Rolle's Theorem guarantees the existence of a point c in the open interval (a, b) where the derivative of the function is zero, implying that the function may have a root or a zero at that point.

Answer

Answer is
B) c ∈ (a, b) such that f'(c) = 0.
Statement for Rolle’s Theorem :
Suppose that a function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b). Then if f(a)=f(b), then there exists at least one point c in the open interval (a,b) for which f′(c)=0.

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