Lesson: The Mean Value Theorem

In this lesson, we will learn how to use the mean value theorem.

Sample Question Videos

  • 01:25

Worksheet: The Mean Value Theorem • 4 Questions • 1 Video

Q1:

Nader is not convinced that the mean value theorem is true because, he says, the function has the property that if we take and , we have , and yet there is no point where . What is his error?

Q2:

Mariam is not convinced that the mean value theorem is true because, she says, the function is certainly differentiable on . But if we take and , we have , and yet there is no point where . What is her error?

Q3:

Consider the result: If 𝑓 is differentiable on an interval 𝐼 and 𝑓 ( π‘₯ ) = 0  , then 𝑓 ( π‘₯ ) = 𝑐 , a constant, for all π‘₯ ∈ 𝐼 .

Which of the following statements says exactly the same thing as the constant function result?

If 𝑓 is differentiable on an interval 𝐼 and not constant, we get points π‘Ž , 𝑏 ∈ 𝐼 with 𝑓 ( π‘Ž ) β‰  𝑓 ( 𝑏 ) . How does this show that 𝑓 β€² ( 𝑐 ) β‰  0 at some point 𝑐 ∈ 𝐼 ?

Q4:

Consider the statement that if 𝑓 is a differentiable function on an interval 𝐼 and 𝑓 β€² ( π‘₯ ) > 0 there, then 𝑓 is strictly increasing on 𝐼 .

Which of the following statements is equivalent to the above?

What does it mean for a function 𝑓 to not be strictly increasing on the interval 𝐼 ?

Using the equivalent statement to the main result, how can you use the mean value theorem to prove the equivalent statement?
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