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# 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  ŌćÆ f (x 1 )Ōēż f (x 2 ) for all x 1 , x2  Ōłł  |a , ]

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

x1 < x2  ŌćÆ f (x 1 )< f (x 2 ) for all x 1 , x2  Ōłł  [a ,b] ## (ii)  Decreasing function

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

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

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

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

A function is said to be monotonic function if it is either an increasing function or a 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 ŌĆ▓ (x ) Ōēź 0

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

### Remarks:

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

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

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

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