What does monotonicity mean?
A function is said to be monotonically increasing (or non-decreasing) if its values are only rising and never falling with increasing values of ( with ). Likewise, it is said to be monotonically decreasing (or non-increasing) if its values are only falling and never rising ( with ).
It is strictly increasing if values always become larger and cannot be constant ( with ). It is strictly decreasing when it is only falling without being constant ( with ).
In order to find the monotonicity of a function, we analyse its first derivative . The derivative is positive at a point if the function is rising and negative if it is falling at this point. The root of the derivative is a point at which the function is neither increasing nor decreasing. If the derivative has at least one root, the entire function cannot be strictly increasing or strictly decreasing. So if no root can be found, we can evaluate the derivative at any point to determine that the function is strictly increasing if it is positive, or strictly decreasing if it is negative. If it has one or more roots, we can still determine its monotonicity in the intervals between the roots.
To find the intervals of the function in which it is rising or falling, we first find the roots of the derivative. At these points the function can change from falling to rising and vice-versa. If the derivative is positive on the left of this point and negative on the right, we know that it is changing from rising to falling. Thus, we can look at a value of the derivative immediately left and immediately right of the root. Another way is to use the second derivative and determine whether it is positive or negative at this point.
Consider the following function:
First, we find its first derivative:
Next, we find the roots of the derivative:
So we know that the monotonicity changes at and . To find how is changes, we evaluate the derivative at three points, left of , between and and finally right of :
means that the function is increasing as it approaches the root of the derivative from the left. It then changes and decreases (as found by ) until it reaches the other root where it changes again and increases from there on (as found by ). The following graph illustrates this. The red line is the function and the purple line is the derivative :
Thus is monotonically increasing for , monotonically decreasing for and monotonically increasing for .
Alternatively, we can determine the monotonicity using the second derivative and evaluate it at the roots of the first derivative that we found already (i.e. at and ). If the second derivative is positive, the monotonicity changed from falling to rising and if it is negative it changes from rising to falling.
Using this method, we arrive at the same conclusion, namely that the function changed from rising to falling at and changed from falling to rising at .