Increasing and Decreasing Functions

This section explains how derivative can be used to check whether a function is increasing, decreasing or neither increasing nor decreasing in its domain.

Let f be a function defined on an interval I and let x₁ and x₂ be any two points on I.

(i) f is said to be increasing in the interval I,

Example:

Define

The graph of the function is as follows:

f(x) is increasing, because

Since 2 - 1 < 3 - 1

(ii) f is said to strictly increasing in the interval I if

For example,

Let x₁ < x₂

f(x) is strictly increasing function.

(iii) f(x) is said to be decreasing function if for x₁, x₂ I

For example,

f(x) = 1 - x for 0 < x < 1

= 0 for 0 x <2

= 2 - x for x 2

is a decreasing function.

(iv) f (x) is said to be strictly decreasing on an interval I if for

For example,

is strictly decreasing function.

Theorem 1:

Let f be continuous on [a, b] and differentiable on the open interval (a, b). Then

(a) f is increasing on [a, b] if f '(x) > 0 for each x (a, b)

(b) f is decreasing on [a, b] if f '(x) < 0 for each x (a, b)

This theorem can be proved by using Mean Value Theorem. We shall prove the theorem after learning Mean Value Theorem.

This theorem is applied in various problems to check whether a function is increasing or decreasing.

Working Rule to Check Whether a Differentable Function is Increasing or Decresing

(1) Let the given function be f (x) on the real number line R.

(2) Differentiate the function f(x) with respect to x and equate it to zero i.e., put f '(x) = 0. Solve for x. These values of x which satisfy f '(x) = 0 are called Critical values of the function

(3) Arrange these Critical values in ascending order and partition the domain of f (x) into various intervals, using the Critical values.

(4) Check the sign of f '(x) in each open intervals.

(5) If f '(x) > 0 in a particular interval, then the function is increasing in that particular interval.

If f '(x) < 0 in a particular interval, then the function is decreasing in that particular interval.

Example:

Find the intervals on which the function

(a) increasing (b) decreasing

Differentiating the function, we have

The critical values in ascending order are -1, 1. We divide the Real numbers into the intervals

= - ve

Since f '(x) < 0, the function is decreasing in the interval _.

Since f '(x) > 0 in the interval (-1, 1), the function is increasing in this interval.

= - ve

Since f '(x) < 0, f(x) is decreasing in the interval (1, ).