Introduction To Monotonicity
In this section, we turn our attention to the increasing / decreasing nature of functions and how the concept of derivatives can help us in determining this nature.
Consider a function represented by the following graph:
For two different input arguments \({x_1}\;and\;{x_2},\) where \({x_1} < {x_2}, \, {y_1} = f\left( {{x_1}} \right)\) will always be less than \({y_2} = f\left( {{x_2}} \right)\) .
That is,
\[{x_1} < {x_2}{\rm{ \;\;\; implies \;\;\; }}f\left( {{x_1}} \right) < f\left( {{x_2}} \right)\]
Such a function is called a strictly increasing function or a monotonically increasing function (The word ‘monotonically’ apparently has its origin in the word monotonous; for example, a monotonous routine is one in which one follows the same routine repeatedly or continuously; similarly a monotonically increasing function is one that increases continuously).
Now, consider \(\begin{align}f\left( x \right) = \left[ x \right].\end{align}\) For this function
\({x_1} < {x_2}\;\text{does not always imply}f\left( {{x_1}} \right) < f\left( {{x_2}} \right)\)
However, \({x_1} < {x_2}\) does imply \(f\left( {{x_1}} \right) \le f\left( {{x_2}} \right)\)
In other words, \(f\left( x \right) = \left[ x \right]\) is not strictly (or monotonically) increasing. It will nevertheless be termed increasing.
Now consider a function represented by the following graph:
For two different input arguments \({x_1}\;and\;{x_2},\) where \({x_1} < {x_2},{y_1} = f\left( {{x_1}} \right)\) will always be greater than \({y_2} = f\left( {{x_2}} \right)\) .
That is,
\[{x_1} < {x_2} \Rightarrow f\left( {{x_1}} \right) > f\left( {{x_2}} \right)\]
Such a function is called a strictly decreasing function or a monotonically decreasing function.
\(\begin{align}\;\;\text{Now consider}\;\;&f\left( x \right) =  \left[ x \right] . \text{For this function}\\\\&{x_1} < {x_2}\;\text{does not imply} \;f\left( {{x_1}} \right) > f\left( {{x_2}} \right)\\\\However, \quad &{x_1} < {x_2} \qquad \qquad \Rightarrow \qquad f\left( {{x_1}} \right) \ge f\left( {{x_2}} \right)\end{align}\)
Therefore, \(f\left( x \right) =  \left[ x \right]\) is not strictly decreasing. It would only be termed decreasing.
The following table lists down a few examples of functions and their behaviour in different intervals. You are urged to verify all the assertions listed on your own.
Function 

Behaviour 
\(f\left( x \right) = x\) 
: 
Strictly increasing on \(\mathbb{R}\) 
\(f\left( x \right) = {x^2}\) 
: 
Strictly decreasing on \(\left( {  \infty ,0} \right]\) Strictly increasing on \(\left[ {0,\infty } \right)\) 
\(f\left( x \right) = \sqrt x \) 
: 
Strictly increasing on \(\left[ {0,\infty } \right)\) 
\(f\left( x \right) = {x^3}\) 
: 
Strictly increasing on \(\mathbb{R}\) 
\(f\left( x \right) = \left x \right\) 
: 
Strictly decreasing on \(\left( {  \infty ,0} \right]\) Strictly increasing on \(\left[ {0,\infty } \right)\) 
\(\begin{align}f\left( x \right) = \frac{1}{x}\end{align}\) 
: 
Neither increasing nor decreasing on \(\mathbb{R} .\) Strictly decreasing on \(\left( {  \infty ,0} \right)\) Strictly decreasing on \(\left( {0,\infty } \right)\) 
\(f\left( x \right) = \left[ x \right]\) 
: 
Increasing on \(\mathbb{R}\) 
\(f\left( x \right) = \left\{ x \right\}\) 
: 
Neither increasing nor decreasing on \(\mathbb{R}.\) However, strictly increasing on \(\left[ {n,n + 1} \right)\) where \(n \in \mathbb{Z}\) 
\(f\left( x \right) = \sin x\) 
: 
Neither increasing nor decreasing on\(\mathbb{R}.\) Strictly increasing on \(\begin{align}[(\,2n  \frac{1}{2}\,)\pi ,(2n + \frac{1}{2})\pi ];n \in \mathbb{Z}\end{align}\) Strictly decreasing on \(\begin{align}[(2n + \frac{1}{2})\pi ,(2n + \frac{3}{2})\pi ];n \in \mathbb{Z}\end{align}\) 
\(f\left( x \right) = \cos x\) 
: 
Neither increasing nor decreasing on \(\mathbb{R}.\) Strictly increasing on \([(2n  1)\pi ,2n\pi ];n \in \mathbb{Z}\) Strictly decreasing on \([2n\pi ,(2n + 1)\pi ];n \in \mathbb{Z}\) 
\(f\left( x \right) = \tan x\) 
: 
Neither increasing nor decreasing on \(\mathbb{R}.\) Strictly increasing on \(\begin{align}\left( {\left( {n  \frac{1}{2}} \right)\pi ,\left( {n + \frac{1}{2}} \right)\pi } \right);n \in \mathbb{Z}\end{align}\) 
\(f\left( x \right) = {e^x}\) 
: 
Strictly increasing on \(\mathbb{R}\) 
\(f\left( x \right) = {e^{  x}}\) 
: 
Strictly decreasing on \(\mathbb{R}\) 
\(f\left( x \right) = \ln x\) 
: 
Strictly increasing on \(\left( {0,\infty } \right)\) 