Question

1. Suppose that ?(?) is differentiable on the interval [1,6] and that ?(1)=1 and ?(6)=6.
a. [5] Sketch a possible graph of ? and the Mean Value Theorem applied to ? on the interval [1,6], including the relevant secant and tangent lines. This should look as is done in the textbook Ch 3.2, and as we did in lecture/recitation.
b. [5] State the conclusion of the MVT about the function ? on the interval [1,6].
c. [3] Now suppose that ? is a differentiable function such that ?′(?)≠1 for all ?. Using what you learned about ? in parts a and b, what can you conclude that ?(?) must not be able to do? Specifically, what cannot be true about the values of ?(1) and ?(6)? Explain.
d. [2] Further, if ? is a differentiable function such that ?′(?)≠1 for all ?, what cannot be true about the values of ?(?) and ?(?), for any ? and ?? Explain.

Answer 1