Question

A friend runs up to you and excitedly explains that she has found that she has found a function g with the following properties:

g is continuous on [0,1] and differentiable on (0,1)
g(0)=1 and g(1)=5
g'(x) is less than or equal to 3 for all x in (0,1)

Explain why you doubt your friend's claim.

Answer 1

Given the start point of (0,1) and (1,5), the average slope is (5-1)/(1-0) = 4. So the finding that g'(x) for the interval is <=3 cannot be true. It might be for a small segment but not the entire interval.