Monotonic (Increasing and Decreasing) Functions
Monotonic functions are functions on real numbers which are either always increasing or always decreasing. Monotonic is a collective name for increasing, strictly increasing, decreasing and strictly decreasing functions.
These can be defined as follows,
if for all x1 < x2
- f(x1) ≤ f(x2) then f is increasing
- f(x1 ) < f(x1 ) then f is strictly increasing
- f(x1 ) ≥ f(x2) the f is decreasing
- f(x1) < f(x2) then f is strictly decreasing
Monotonicity and Derivatives
If a function f(x) is increasing then what we mean is that the slope is always positive, so if f is continuous we can relate the the properties of increasing and decreasing to the derivative as shown in the table below.
| Increasing/Decreasing | condition of f’(x) |
|---|---|
| Increasing | f’(x) ≥ 0 |
| Strictly Increasing | f’(x) > 0 |
| Decreasing | f’(x)≤ 0 |
| Strictly Decreasing | f’(x) <0 |
Its important to note that these rules only work if the function is continuous, for example consider f(x) =1/x, which is discontinuous at 0.
We can differentiate it to get f’(x) = -1/x2 which we know is always negative (because the squared term is always positive) so we would expect it to be a decreasing function. However if we consider two point either side o, 1 and -1 say we find that f is not a decreasing function because whilst -1 < 1, -1/x < 1/x contrary to our definition of decreasing
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