JEE Mathematics/ Lakshmikanta Satapathy/ Application of Derivative part 3/ Understanding Increasing and Decreasing Functions using graphs and the first derivative

1. Physics Helpline
L K Satapathy
Increasing / Decreasing Functions 1

2. Physics Helpline
L K Satapathy
Application of Derivative 3
Increasing and Decreasing Functions
Increasing and Decreasing functions
Consider a real valued function f ( x )
and an interval ( I ) contained in the domain of f ( x )
Then f is increasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
f is strictly increasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
f is decreasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
f is strictly decreasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
1x 2x
1( )f x
2( )f x
x
( )f x

3. Physics Helpline
L K Satapathy
Application of Derivative 3
Increasing and Decreasing Functions
(1) f is increasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
(2) f is strictly increasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
As x increases ,
f (x) either increases [ x1 to x2 ] and [ x3 to x4 ]
or remains constant [ x2 to x3 ]
As x increases , f (x) always increases
1x 2x 3x 4x x
( )f x
1x 2x x
( )f x

4. Physics Helpline
L K Satapathy
Application of Derivative 3
Increasing and Decreasing Functions
(3) f is decreasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
(4) f is strictly decreasing in I , if
1 2 1 2 1 2( ) ( ) ,x x f x f x for all x x I   
As x increases ,
f (x) either decreases [ x1 to x2 ] and [ x3 to x4 ]
or remains constant [ x2 to x3 ]
As x increases , f (x) always decreases
1x 2x 3x 4x x
( )f x
1x 2x x
( )f x

5. Physics Helpline
L K Satapathy
Application of Derivative 3
Increasing and Decreasing Functions
1st derivative method to determine increasing and decreasing functions
Let f be a function continuous in [a , b] and differentiable in (a , b) , then
( i ) f is strictly increasing in [a , b] if f(x)  0 for each x (a , b)
( ii ) f is strictly decreasing in [a , b] if f(x)  0 for each x (a , b)
( iii ) f is a constant function in [a , b] if f(x) = 0 for each x (a , b)
f is strictly increasing in [a , b]  f(x)  0
f is strictly decreasing in [c , d]  f(x)  0
f is a constant function in [b , c]  f(x) = 0
In the figure
a b c d

6. Physics Helpline
L K Satapathy
Application of Derivative 3
Increasing and Decreasing Functions
Procedure
Differentiate the given function f (x). Let f (x) = (x – a)(x – b)
Solve the equation f (x) = 0  (x – a)(x – b) = 0  x = a or x = b
a b
( , )a ( , )a b ( , )b 
( , )x a   f (x) = (x – a) . (x – b) = (–).(–) = +ve (increasing)
( , )x a b  f (x) = (x – a) . (x – b) = (+).(–) = –ve (decreasing)
( , )x b   f (x) = (x – a) (x – b) = (+).(+) = +ve (increasing)
Points x = a and x = b divide the number line into three intervals as shown
Rule : Factors on right of a point are – ve
Factors on left of a point are +ve

7. Physics Helpline
L K Satapathy
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