How do you use Rolle's Theorem to Prove the Mean Value Theorem?

In summary, the Mean Value Theorem can be proved by applying Rolle's Theorem to a special case of MVT, where the function satisfies the conditions of both Rolle's and MVT. This special case involves a function that can be written in the form of a matrix, where the first and second columns have fixed values of 1 and a/b respectively, and the third column has varying values from a to b. By applying Rolle's theorem to this function, it can be shown that there exists a point c between a and b where the derivative is 0, proving the Mean Value Theorem.
  • #1
nuadre
6
0

Homework Statement



Assuming Rolle's Theorem, Prove the Mean Value Theorem.


Homework Equations



-

The Attempt at a Solution



I know these definitions:

Rolle's Theorem:
If y=f(x) is continuous on all points [a,b] and differentiable on all interior points (a,b),
and if f(a) = f(b)
then there is at least one point c such that f '(c)=0

Mean Value Theorem:
y:f(x) continuous on [a,b], differentiable on all (a,b) then one point such that

f(b) - f(a) / b-a = f ' (c)

------------------------------

Do i substitute values in for a,b,c to try to prove? I'm so horribly confused :(
 
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  • #2
Of course in the MVT it's not necessarily true that f(a) = f(b). Can you subtract a natural function from f(x), let's call this h(x), so that g(x) = f(x) - h(x) satisfies g(a) = g(b)? Then, you will be able to apply Rolle's theorem.
 
  • #3
hi Ted! Thanks for your reply, I couldn't get my head around this.

Can I use values of a= -1 , b= 1, c=0 and sub these in for a function
y= x^2 ?
y' = 2x
so

y(-1) = f(a)
y(1) = f(b)
y'(0) =0
therefore f(b) - f(a) / b-a = f ' (c) is equal to
1-1 / 1+1 =0 =f'(c) ?

Am I on the right track? How would I word this for an answer though ha.
 
  • #4
No, the MVT is true for all intervals [a,b] and all functions f(x). Substituting values for a and b and a special form f(x) = x^2 is only a special case of MVT. It doesn't prove it in general. First, do you understand why Rolle's theorem is a special case of MVT?
 
  • #5
Proof of Rolle's Theorem:
"From the extreme value theorem, the function attains its extreme values on [a,b]. If it attains them both at a and b, then the function is constant, and so has zero derivative everywhere. If it attains either of them at an interior point, then by the extreme value derivative theorem the derivative at that point is zero."
-------------------------------------
"From the extreme value theorem, the function attains its extreme values on [a,b]. If it attains them both at a and b"

Is this saying it obtains both its global max/global min at both points a and b? So I would be visualising a straight line parallel to the x axis?

"If it attains either of them at an interior point, then by the extreme value derivative theorem the derivative at that point is zero"

I'm now visualising a point c, situated between a and b. If there is a global max/min here then that means the derivative of c = 0?
 
  • #6
Yes, but explain how it is a special case of MVT. That is, given MVT, why is it almost immediately obvious that Rolle's theorem is true.
 
  • #7
it's a special case of MVT because at every point between a,b inclusive, the derivative is 0 (As it is a constant straight line parallel to x axis)? Which proves that rolle's theorem is true?
 
  • #8
Rolle's theorem doesn't only apply to straight, horizontal lines.
 
  • #9
nuadre said:

Homework Statement



Assuming Rolle's Theorem, Prove the Mean Value Theorem.

Try applying Rolle's theorem to

[tex]F(x) = \left| \begin{array}{ccc}
1 & a & f(a)\\
1 & b & f(b)\\
1 & x & f(x)
\end{array}\right|[/tex]
 

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 values of the function 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. How is Rolle's Theorem used to prove the Mean Value Theorem?

Rolle's Theorem is used to prove the Mean Value Theorem by providing the necessary conditions for the Mean Value Theorem to hold. It shows that if a function satisfies the conditions of Rolle's Theorem, then it must also satisfy the conditions of the Mean Value Theorem.

3. Can Rolle's Theorem be applied to any function?

No, Rolle's Theorem can only be applied to functions that are continuous on a closed interval and differentiable on the open interval. If a function does not meet these conditions, then Rolle's Theorem cannot be used to prove the Mean Value Theorem.

4. What is the significance of the Mean Value Theorem?

The Mean Value Theorem is significant because it provides a way to find the average rate of change of a function over an interval. This can be useful in various applications, such as determining average velocity or average acceleration.

5. Are there any other theorems related to Rolle's Theorem and the Mean Value Theorem?

Yes, there are several other theorems that are related to Rolle's Theorem and the Mean Value Theorem, such as the Intermediate Value Theorem and the First and Second Derivative Tests. These theorems all build upon the concepts of continuity, differentiability, and the relationship between a function and its derivative.

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