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.

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

*x*_{1} < *x*_{2} â‡’ *f*
(*x* _{1} )â‰¤ *f* (*x* _{2} ) for all *x* _{1} , *x*_{2} âˆˆ |*a* , ]

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

*x*_{1} < *x*_{2} â‡’ *f*
(*x* _{1} )< *f* (*x* _{2} ) for all *x* _{1} , *x*_{2} âˆˆ [*a* ,*b*]

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

*x*_{1} < *x*_{2} â‡’ *f*
(*x* _{1} )â‰¥ *f* (*x* _{2} ) for all *x* _{1} , *x*_{2} âˆˆ [*a* ,*b*]

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

*x*_{1}* *<* x*_{2}* *â‡’* f *(*x *_{1}* *)>* f *(*x *_{2}* *)* *for all* x *_{1}* *,* x*_{2}* *âˆˆ* *[*a *,*b*]

**NOTE**

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

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 *â€²* *(*x *)* *â‰¥* *0

(ii) *f*(*x*) is decreasing in [*a*,*
b*] if* f *â€²* *(*x *)* *â‰¤* *0

(i) *f*(*x*)
is strictly increasing in (*a*,*b*) if*
f *â€²* *(*x *)* *>* *0* *for every* x *âˆˆ* *(*a *,*b*)

(ii) *f*(*x*)
is strictly decreasing in (*a*,*b*) if*
f *â€²* *(*x *)* *<* *0* *for every* x *âˆˆ* *(*a *,*b*)

(iii) *f*(*x*) is said to be a constant
function if* f *â€²* *(*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

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