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Maxima and minima - Applications of Differentiation - Increasing and decreasing functions | 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation

Chapter: 11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation

Increasing and decreasing functions

Before learning the concept of maxima and minima, we will study the nature of the curve of a given function using derivative.

Increasing and decreasing functions

Before learning the concept of maxima and minima, we will study the nature of the curve of a given function using derivative.

(i) Increasing function

A function f(x) is said to be increasing function in the interval [a,b] if

x1 < x2  f (x 1 )≤ f (x 2 ) for all x 1 , x2    |a , ]

A function f(x) is said to be strictly increasing in [a,b] if

x1 < x2  f (x 1 )< f (x 2 ) for all x 1 , x2    [a ,b]


(ii)  Decreasing function

A function f(x) is said to be decreasing function in [a, b] if

x1 < x2  f (x 1 )≥ f (x 2 ) for all x 1 , x2    [a ,b]

A function f(x) is said  to be strictly decreasing function in [a,b] if x

x1 < x2 f (x 1 )> f (x 2 ) for all x 1 , x2 [a ,b]


NOTE

A function is said to be monotonic function if it is either an increasing function or a decreasing function.

Derivative test for increasing and decreasing function

Theorem:6.1 (Without Proof)

Let f(x) be a continuous function on [a,b] and differentiable on the open interval (a,b), then

(i) f(x) is increasing in [a, b] if f (x ) 0

(ii) f(x) is decreasing in [a, b] if f (x ) 0

Remarks:

(i) f(x) is strictly increasing in (a,b) if f (x ) > 0 for every x (a ,b)

(ii) f(x) is strictly decreasing in (a,b) if f (x ) < 0 for every x (a ,b)

(iii) f(x) is said to be a constant function if f (x ) = 0

 

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11th Business Mathematics and Statistics(EMS) : Chapter 6 : Applications of Differentiation


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