Application of Derivatives


   
 
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, ).
 
 
     
   
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