5. Suppose that I' exists and is continuous on a nonempty, open interval

(a,

b) with I'(x) =I- 0 for all x E

(a, b).

(a) Prove that I is I-Ion

(a,

b) and takes

(a,

b) onto some open interval

(c, d).

(b) Show that (1-1)' exists and is continuous on

(c, d).

(c) Using the function I (x) = x3 , show that

(b) is false if the assumption I' (x) =Io fails to hold for some x E

(a, b).

(d) Sketch the graphs of y = tan x and y = arctan x to see that c and d in part

(b) might be infinite.