Suppose that f ′ exists and is continuous on a nonempty, open interval (a, b) with f ′(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 be infinite.
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