MONOTONIC FUNCTIONS

Guys..!! Here's a very quick way to learn about monotonic functions, their properties, to find the maxima, minima and their properties etc.

Monotonic function :
A function f(x ) is said to be monotonic on an interval (a,b) if it is either increasing or decreasing  (a,b)

Definition 1: A function f(x) is said to be increasing (decreasing) at a point  containing, if there is an interval  such that f(x) is increasing (decreasing) On .

Definition 2:  A function f(x) is said to be increasing (decreasing ) on [a,b] if it is increasing (decreasing) on (a,b) and it is also increasing (decreasing ) on [a,b] and its is also increasing (decreasing ) at x =a and x=b .

Necessary and sufficient condition Montonicity of functions:

Necessary condition : Let f(x) be a differentiable function  defined on (a,b) then  or  according as f(x) is increasing or decreasing on (a,b)

If  f(x) is an  increasing function on (a,b) then tangent at every point on curve y =f(x) makes an acute angle in the +ve direction of x-axis.

 or .

Similarly for decreasing  function  on (a,b) then the tangent at every point on the curve 

 y =f(x) makes obtuse angle  with direction of x –axis.

Sufficient condition

Let f be a differentiable real function defined on open interval (a,b)

a) If  , then f is increasing on (a,b)

b) If ,then f is decreasing on (a,b)

Properties of Monotonic functions :

i) If f(x)  is  strictly increasing function on an interval [a,b] then exists and also a strictly increasing function

ii) If f(x) is strictly increasing function on an interval [a,b] such that  it is continuous then    is continuous on [f(a) ,f(b)]

iii) If f(x) is continuous on  [a,b] such that f’(c)  then f(x) is monotonically  (strictly increasing function on [a,b])

Maxima and Minima

Maxima : Let f(x) be function with domainthen f(x) is said to attain the maximum value at a  point  if . A is called the point of maximum f(a) is known as Maximum value of  f(x).

Minimum :

Let f(x) be a function with Domain  . Then f(x) is said to attain the minimum value at a point  if . The point a is called the point minimum f(a) is known as Minimum value of f(x).

Local Maxima :  

 A function f(x) is said to attain a local maximum at x=a  if  a neighborhood  of a such that .

In such a case f(a) is called the local maximum value of  f(x) at x =a

Local maximum value of f(x) at x =0

Local minima : A function f(x) is said to attain a local minimum at x =a if  a neighbor hood  of a such that  then f(a) is called local minima of f(x) at x =a.

 The points at which a function attains either the local maximum values or local minimum values are  known as extreme points and both local maximum and minimum  values are called the extreme values of f(x) At the extreme points  of f(x) .

Note :-

   By a local maximum or (Minimum) values of a function at a point x =a mean the greatest or (the least) value in the nbd of point x =a and not  the absolute  minimum . In fact a function may have any number of points of local maximum or ( local minimum) 

 A necessary condition for  f(a) to be an extreme value of function is that  in case it exist.

Points of inflection:

A point of inflection is a point at which a curve is changing concave upwards to concave downwards (or)   vice –versa 

  A curve y  = F(x) has one of its points  and changes  sign as x is increases through x =C.

An arc  y =f(x)  is called concave downward if , at each of  the  points the arc lies below  the tangent 

·         f”(x) <0 for concave downward curve

·         f”(x)>0 for concave upwards curve 

Properties of maxima and minima

I) If f(x) is continuous function   in its domain ,the at least one maxima and one minima  must lies. Between two equal values of x.

II)Maxima and Minima  Occur alternatively, between two maxima  there is one minimum and vice-versa

III) If  or b f’(x) =0 for any one value of x between a and b then f(c) is necessarily the minimum and the least value  or b then f(c) is necessarily the maximum.

Maximum and minimum values of f(x) in closed interval

Let y = f(x) be function  defined on

Steps

 1) find .

2) Put f’(x) =0 and find values of x let  be values of x

3)  Take the maximum and minimum out of the values 

F( a) , .

Therefore, the maximum and minimum values are absolute maximum or minimum values of the function.