What does monotonicity mean?

A function equation is said to be monotonically increasing (or non-decreasing) if its values are only rising and never falling with increasing values of equation (equation with equation). Likewise, it is said to be monotonically decreasing (or non-increasing) if its values are only falling and never rising (equation with equation).

It is strictly increasing if values always become larger and cannot be constant (equation with equation). It is strictly decreasing when it is only falling without being constant (equation with equation).

In order to find the monotonicity of a function, we analyse its first derivative equation. 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 equation 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.

Example

Consider the following function:

    equation

First, we find its first derivative:

    equation

Next, we find the roots of the derivative:

(1)   equation

So we know that the monotonicity changes at equation and equation. To find how is changes, we evaluate the derivative at three points, left of equation, between equation and equation and finally right of equation:

(2)   equation

equation means that the function is increasing as it approaches the root of the derivative equation from the left. It then changes and decreases (as found by equation) until it reaches the other root equation where it changes again and increases from there on (as found by equation). The following graph illustrates this. The red line is the function equation and the purple line is the derivative equation:

monotonicity

Thus equation is monotonically increasing for equation, monotonically decreasing for equation and monotonically increasing for equation.

Alternatively, we can determine the monotonicity using the second derivative equation and evaluate it at the roots of the first derivative that we found already (i.e. at equation and equation). If the second derivative is positive, the monotonicity changed from falling to rising and if it is negative it changes from rising to falling.

(3)   equation

Using this method, we arrive at the same conclusion, namely that the function changed from rising to falling at equation and changed from falling to rising at equation.