# Function Help - Rolle's Theorem or the Mean Value Theorem?

• MHB
• vickon
In summary, using Rolle's Theorem, there exists a point c1 in (0,1) such that f'(c1) = 0 and using it again, there exists a point c2 in (0,c1) such that f''(c2) = 0.
vickon
Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem, and verify that the conditions of
the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?

vickon said:
Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem, and verify that the conditions of
the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?
Yes, you should use Rolle's theorem twice: first for the function $f$ and then for the function $f'$.

vickon said:
Let f be a function twice-differentiable function defined on [0, 1] such that f(0)=0, f′(0)=0, and f(1)=0.
(a) Explain why there is a point c1 in (0,1) such that f′(c1) = 0.
(b) Explain why there is a point c2 in (0,c1) such that f′′(c2) = 0.
If you use a major theorem, then cite the theorem, and verify that the conditions of
the theorem are satisfied.

Should I be using Rolle's Theorem/Mean Value Theorem?

This problem just needs straight forward application of Rolle's Mean Value Theorem as already mentioned by the problem poster.

## 1. What is Rolle's Theorem?

Rolle's Theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, and the function's values at the endpoints of the interval are equal, then there exists at least one point within the interval where the derivative of the function is equal to zero.

## 2. What is the Mean Value Theorem?

The Mean Value Theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the derivative of the function is equal to the average rate of change of the function over the interval.

## 3. What is the difference between Rolle's Theorem and the Mean Value Theorem?

The main difference between Rolle's Theorem and the Mean Value Theorem is that Rolle's Theorem only guarantees the existence of one point where the derivative is equal to zero, while the Mean Value Theorem guarantees the existence of at least one point where the derivative is equal to the average rate of change of the function.

## 4. How are Rolle's Theorem and the Mean Value Theorem used in real-world applications?

Both Rolle's Theorem and the Mean Value Theorem are used in real-world applications to analyze and optimize functions. For example, they can be used in economics to find the optimal production level for a company or in engineering to find the maximum efficiency of a machine.

## 5. Can Rolle's Theorem or the Mean Value Theorem be applied to all functions?

No, Rolle's Theorem and the Mean Value Theorem have certain conditions that must be met in order for them to be applicable. For example, the function must be continuous on a closed interval and differentiable on the open interval. Additionally, the Mean Value Theorem requires that the function's derivative is continuous on the interval. If these conditions are not met, the theorems cannot be applied.

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