Sawyer

Some Thoughts on Examinations (WW Sawyer)

2) Give examples of continuous functions that satisfy the following conditions, or explain why the requirements are impossible.
(a) f (0) = 1, f (1) =0, f'(x) positive for all x.
(b) f (0) = 1, f' (0) = -1, f"(x) always negative, f(x) =0 having no real solution.

Of course, in each part of this question the requirements are impossible. In (a) the function is required to be increasing and yet have f(1) less than f(0). In (b) the graph must have the slope and curvature as shown here, which clearly compels f (x) =0 to have a solution between x=0 and x=l, contrary to the last requirement.