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

Q1:

Madison is not convinced that the mean value theorem is true because, she says, the function π ( π₯ ) = | π₯ | is certainly differentiable on β β { 0 } . But if we take π = β 1 and π = 1 , we have π ( π ) β π ( π ) π β π = 0 , and yet there is no point π₯ where π ( π₯ ) = 0 β² . What is her error?

Q2:

Consider the result: If π is differentiable on an interval πΌ and π ( π₯ ) = 0 β² , 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 ?

Q3:

Mason is not convinced that the mean value theorem is true because, he says, the function π ( π₯ ) = | π₯ | has the property that if we take π = β 2 and π = 2 , we have π ( π ) β π ( π ) π β π = 0 , and yet there is no point π₯ where π β² ( π₯ ) = 0 . What is his error?

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