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

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 π is differentiable on an interval πΌ and π ( π₯ ) = 0 ο , then π ( π₯ ) = π , a constant, for all π₯ β πΌ .

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

If π is differentiable on an interval πΌ and not constant, we get points π , π β πΌ with π ( π ) β π ( π ) . How does this show that π β² ( π ) β 0 at some point π β πΌ ?

Q4:

Consider the statement that if π is a differentiable function on an interval πΌ and π β² ( π₯ ) > 0 there, then π is strictly increasing on πΌ .

Which of the following statements is equivalent to the above?

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

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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