Needed clarification regarding Rolles theorem

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In summary, Rolle's Theorem states that if a function is continuous on a closed interval, differentiable on the open interval, and has the same values at the endpoints, there exists a point within the open interval where the derivative is equal to zero. The first condition is necessary as differentiability does not guarantee continuity on the closed interval. This can be seen with examples like f(x) = 1/x for x in (0,1] and 1 for x = 0.
  • #1
seshikanth
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Hi,
If we consider Rolle's Theorem:
"If [1] f is continuous on [a, b],
[2] differentiable in (a,b), and
[3]f (a) = f (b), then there exists a point c in (a, b) where f'(c) = 0."

In the above theorem, i did not get why condition [1] is needed as we have differentiability => continuity. Can Holes at a and b (removable discontinuity) can be taken care by condition [3]?

Thanks,
 
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  • #2
Try piecewise things. For example,
what about f(x) = 1/x for x in (0,1] and 1 for x = 0? Then "f(0)" is defined (namely, f(0) = 1) and we have f(0) = f(1) but obviously f'(c) is never zero on (0,1)
 
  • #3
Notice that the intervals are different. f is assumed to be differentiable on the open interval (a, b) so it is continuous on that interval but not necessarily on the closed interval [a, b] as l'Hôpital's example shows.
 

1. What is Rolle's Theorem?

Rolle's Theorem is a fundamental theorem in calculus that states that if a real-valued function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) with f(a) = f(b), then there exists at least one point c in the open interval where the derivative of the function is equal to zero.

2. Why is Rolle's Theorem important?

Rolle's Theorem is important because it provides a necessary condition for a function to have a horizontal tangent line. It also serves as the basis for other important theorems in calculus, such as the Mean Value Theorem and the Fundamental Theorem of Calculus.

3. How is Rolle's Theorem applied in real-world situations?

Rolle's Theorem can be applied in real-world situations to find the maximum or minimum values of a function. It can also be used to prove the existence of a solution to a given problem, such as finding the roots of an equation or determining the velocity of an object at a specific time.

4. What are the limitations of Rolle's Theorem?

Rolle's Theorem has several limitations. It only applies to functions that are continuous and differentiable on a closed interval. It also requires that the endpoints of the interval have the same function values. Additionally, it does not provide any information about the location of multiple points where the derivative is equal to zero.

5. How is Rolle's Theorem related to the Intermediate Value Theorem?

Rolle's Theorem is a special case of the Intermediate Value Theorem, where the function values at the endpoints are equal. The Intermediate Value Theorem states that if a continuous function has different values at the endpoints of an interval, then it must take on every value in between those endpoints at least once. Rolle's Theorem is often used to prove the Intermediate Value Theorem.

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