Maximum and minimum of a function also called as extremum of a function.
The maximum and minimum of a function , Let y=f(x) be a function whose domain is interval “D” i.e $ x \,\in\, D\,$ . In the interval D when does we say a point to be one of extrimum , i.e either maximum and minimum of a function.
There are two types of maximum and minimum of a function
- Local maximum and local minimum
- Global maximum and minimum
What is Local Maximum and Minimum of a function
Local Maximum :::: As the name specify when a function have a value greater than its right and left neighbor value then the point at which it is greater is point of local maximum.
i.e if f( a ) > f ( a – h ) and f ( a ) > f ( a + h ) then point ( a , f ( a ) ) is point of local maximum.
Local Minimum :::: As the name specify if the value of function for an x is less than its left and right neighbor value , Then the point have a local minimum at that point.
if f ( a ) < f ( a – h ) and f ( a ) < f ( a + h ) then point ( a , f ( a ) ) is point of local minimum.
Global Maximum and Global Minimum
The maximum value of function in its domain is called as Global maximum . If f ( a ) $\ge\,$ f ( x ) $\lor\,x\,\in \,D\,$ then f ( a ) is Global maximum of function.
The minimum value of function in its domain is called as Global minimum . If f ( a ) $\le\,$ f ( x ) $\lor\,x\, \in \, D\, $ then f ( a ) is Global minimum of the function.
Thus the Local or Global maximum and minimum of the function are the value of function for some x in the domain of the function.
NOTE : Limiting value of function cannot be considered as either Local Or Global maximum and minimum of function.
Methods to find the Local Maximum and Minimum of a function
Local Maximum and Minimum of a function
If function is a discontinuous function or a discrete valued function , then basic definition of Local maximum and minimum of a function from graph is the only method.
Whereas if a function is continuous and differentiable in its domain the we use the method of derivative .
Let at x = a , f'( a ) = 0 , and f’ ( a – h ).f’ ( a + h ) < 0 then x = a is either point of local maximum and minimum of the function and f ( a ) is either local maximum or minimum of function.
Global Maximum and Minimum of function.
The method used to find Global maximum and minimum of a function is method of derivative.
If f ( x ) is continuous and differentiable function for x $\in$ [ a , b ]
Step 1 : Find first derivative of function and hence find the points where f ‘ ( x ) = 0 . Let f ‘ ( x ) = 0 for x = $ x_1\,,\,x_2\,,\,x_3\,…… $
Step 2 : Find the value of function at all the points where f ‘ ( x ) is zero . Find f ( $x_1$ ) , f ( $x_2$ ) , …….. , also find value of function at extremity of domain at x = a and at x = b , f ( a ) and f ( b ).
Define a set S = { f ( a ) , f ( $x_1$ ) , f ( $x_2$ ) , ……. , f ( b ) } . Maximum from set S is Global maximum and minimum from set S is Global minimum.
Where as if Domain of function f is ( a , b ) i.e open interval
Step 1 : Find first derivative of function and hence find the points where f ‘ ( x ) = 0 . Let f ‘ ( x ) = 0 for x = $ x_1\,,\,x_2\,,\,x_3\,…… $
Step 2 : Find the value of function at all the points where f ‘ ( x ) is zero . Find f ( $x_1$ ) , f ( $x_2$ ) , …….. , also find limiting value of function at extremity of domain at x = a and at x = b . Let $l_1\,=\,\lim_{x\to a^{+}}f ( \,x\, ) \,\,\, , \,\, l_2\,=\,\lim_{x\to b^{-}}f(\,x\,)$ .
Define a set S = { f ( $x_1$ ) , f ( $x_2$ ) , ……. } . Let M = Maximum from set S and m = minimum from set S.
If maximum{$\,l_1\,,\,l_2\,$} > M then Global maximum of function does not exist else M is Global maximum value of function
If minimum{$\,l_1\,,\,l_2\,$} < m then Global minimum of function does not exist else m is Global minimum value of function .