Monotonic Function

A function is said to be monotonic on an interval 'I' if it is either an increasing function or a decreasing function on I.

Properties of monotonic functions:

(i) If f(x) is a strictly increasing function on an interval [a, b],
then f –1 exists and it is also a strictly increasing function.
(ii) If f(x) is a strictly increasing function on an interval [a, b] such that it is continuous,
then f –1 is continuous on [f(a), f(b)].
(iii) If f(x) is continuous on [a, b] such that f '(c) ≤ 0
for each c ∈ (a, b),
then f(x) is monotonically decreasing function on [a, b].
If f '(c) < 0, then f(x) is strictly decreasing function.
(iv) If f(x) is continuous on [a, b] such that f '(c) ≥ 0
for each c ∈ (a, b), then f(x) is monotonically increasing function on [a, b].
If f '(c) > 0, then f(x) is strictly increasing function.
Many text books assert the truth of following two statements (v and vi) without proper validation:
Actually the above two statements not true. This is explained with counter examples under:
(v) If f(x) and g(x) are monotonically increasing functions on [a, b],
then (gof)x is a monotonically increasing function on [a, b].



(vi) If one of the two functions f(x), g(x) is strictly increasing and other a strictly decreasing,
then (gof)x is strictly decreasing on [a, b].