Maxima and Minima Notes.

let's understand in an informal way:

Suppose a function f(x) is given and some boundary value is associated with it for example:

f(x)= 4x^{3} + 6x^{2} +17 and boundary value is {0, 6}

 The only questions that can be asked are:-

1. Find the points of local maxima and minima.

2. Find the maximum or minimum value of f(x). 

Critical points:-

let us suppose a function f(x) is given and a point a is defined as a critical point if

f'(a) =0.

Sometimes it is also called as stationary points.

Method to find Maxima and Minima:-

let us say our function  is represented as f(x)

1. first we  have to find out f'(x) 

2. Now find stationary points by f'(x) =0. you may get one or more critical points. Suppose 'a' is our critical point.

3. Now find out f''(x)

Case1: If f''(a) < 0 then  'a' is called point of maxima and f(x) has maximum value at this point.

Case 2: if f''(a) >0 then 'a' is called the point of minima and f(x) has a minimum value at this point.

Case 3: if f''(a) =0 then we need to derivative test.


Derivative test.

Informally, when we get even derivative like f''(a) =0 then we have to apply double derivate test.

We need to find odd derivate f'''(x) 

Case 1: if f'''(a) = 0  ,then find further even derivate like f''''(x).

Case 2: if f'''(a) ! = 0, then the given function doesn't have any maxima or minima.