# Maximum and Minimum of a function

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

1. Local maximum and local minimum
2. 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 .