Suppose that f' exists and is continuous on a nonempty, open interval (a, b) with f(x)

Question:

Suppose that f' exists and is continuous on a nonempty, open interval (a, b) with fʹ(x) ≠ 0 for all x ∈ (a, b).
a) Prove that f is 1-1 on (a, b) and takes (a, b) onto some open interval (c, d).
b) Show that f-1 ∈ C1 (c, d).
c) Using the function f(x) = x3, show that b) is false if the assumption f'(x) ≠ 0 fails to hold for some x ∈ (a, b).
d) Sketch the graphs of y = tan x and y = arctan x to see that c and d in part b) might bev infinite.