5.5 Increasing and decreasing intervals

A function is strictly increasing, strictly decreasing, non-decreasing and non-increasing in a suitably selected interval in the domain of the function. We have seen that a linear algebraic function maintains order of change throughout its domain. The order of change, however, may not be maintained for higher degree algebraic and other functions in its domain. We shall, therefore, determine monotonic nature in sub-intervals or domain as the case be.

One of the fundamental ways to determine nature of function is by comparing function values corresponding to two independent values ( $x_1$ and $x_2$ ). This technique to determine nature of function works for linear and some simple function forms and is not useful for functions more complex in nature. In this module, we shall develop an algorithm based on derivative of function for determining nature of function in different intevals.

In the discussion about monotonic function in earlier module, we observed that order of change in function values is related to sign of the derivative of function. The task of finding increasing and decreasing intervals is, therefore, about finding sign of derivative of function in different intervals and determining points or intervals where derivative turns zero.

Nature of function and intervals

The steps for determining intervals are given as under :

1: Determine derivative of given function i.e. f’(x).

2: Determine sign of derivative in different intervals.

3: Determine monotonic nature of function in accordance with following categorization :

$$f\prime \left(x\right)\ge 0:\text{equality holding for points only – strictly increasing interval}$$ $$f\prime \left(x\right)\ge 0:\text{equality holding for subsections also – non-decreasing or increasing interval}$$ $$f\prime \left(x\right)\le 0:\text{equality holding for points only – strictly decreasing interval}$$ $$f\prime \left(x\right)\le 0:\text{equality holding for subsections also – non-increasing or decreasing interval}$$

5: The interval is open “( )” at end points, if function is not continuous at end points. However, interval is close “[]” at end points, if function is continuous at end points.

In order to illustrate the steps, we consider a function,

$$f\left(x\right)=x^2-x$$

Its first derivative is :

$$f\prime \left(x\right)=2x-1$$

Here, critical point is 1/2. First derivative, f’(x), is positive for x>1/2 and negative for x<1/2. The signs of derivative are strict inequalities. It means that function is either strictly increasing or strictly decreasing in the open intervals. We know that infinity end is an open end. But, function is continuous in the given interval. Hence, we can include end point x=1/2. Further, since derivative is zero at x=1/2 i.e. at a single point, function remains strictly increasing or decreasing.

$$\text{Strictly increasing interval}=[-\infty ,\frac{1}{2}]$$ $$\text{Strictly decreasing interval}=[\frac{1}{2},\infty ]$$

Algebraic functions

Derivative of algebraic function is also algebraic. In order to determine sign of derivative, we use sign scheme or wavy curve method, wherever expressions in derivative can be factorized.