Introduction To Monotonicity

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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 \(\left( {0,\infty } \right)\)

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Applications of Derivatives
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Applications of Derivatives
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