Mason is not convinced that the mean value theorem is true because he says, the function 𝑓 of 𝑥 equals the absolute value of 𝑥 has the property that if we take 𝑎 equals negative two and 𝑏 equals two, we have 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus 𝑎 equals zero, and yet there is no point 𝑥 where 𝑓 prime of 𝑥 equals zero. What is his error?
To use the mean value theorem, two things must be true. 𝑓 of 𝑥 must be continuous over 𝑎, 𝑏. When we consider Mason’s function 𝑓 of 𝑥 equals the absolute value of 𝑥, this statement is true. 𝑓 of 𝑥 is continuous over 𝑎, 𝑏. It must also be true that our 𝑓 of 𝑥 is differentiable over 𝑎, 𝑏.
And for Mason, 𝑓 prime of the absolute value of 𝑥 equals 𝑥 over the square root of 𝑥 squared and 𝑥 cannot be equal to zero. The function 𝑓 of 𝑥 equals the absolute value of 𝑥 is not differentiable over the interval 𝑎, 𝑏. And that means the mean value theorem cannot apply here.