Find The Local Maximum And Minimum Values
Find The Local Maximum And Minimum Values
In mathematics we Find the Local Maximum and Minimum values for the functions. There are
points in the domain of a function where the function does not attain the greatest value but the
values at these points are greater than or less than the values of the function at the
neighboring points.
Such points are called the local minimum and local maximum point. Let us discuss about the
local minimum and local maximum values for the function.
Local maximum: - A function f ( x ) reaches a local maximum at x = a if there is a region ( a -,
a + ) of a such that
f ( x ) < f ( a ) for al x ( a - , a + ), x a
or f ( x ) - f ( a ) < 0 for al x ( a - , a + ), x a
In this situation f ( a ) is called the local maximum value of f ( x ) at x = a.
Local minimum : - A function f ( x ) reaches a local maximum at x = a if there is a region ( a -,
a + ) of a such that
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f ( x ) > f ( a ) for al x ( a - , a + ), x a
or f ( x ) - f ( a ) > 0 for al x ( a - , a + ), x a
In this situation f ( a ) is called the local minimum value 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 the extreme points.
Let's take an example to find al the points of local maximum and local minimum of the
function.
f ( x ) = x3 - 6x2 + 9x - 8
Solution : - Let y = f ( x ) = x3 - 6x2 + 9x - 8. Then,
dy / dx = f' ( x ) = 3x - 12x + 9x = 3 ( x2 - 4x + 3 )
For a local maximum and local minimum, we have
dy / dx = 0 => 3 ( x2 - 4x + 3 ) = 0 => x = 1, 3.
We have to examine whether these points are points of local maximum or local minimum or
neither of them
We have dy / dx 3 ( x - 1 ) ( x - 3 )
The change in signs of dy / dx for different values of x are shown in figure.

Clearly, dy / dx changes sign from positive to negative as increase through 1.
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Find the Local Maximum and Minimum valuesp: initial; background-color: white; background-
position: initial initial; background-repeat: initial initial; "> x = 1 is a point of a local maximum.
Also, dy / dx changes sign from negative to positive as x increases through 3.
So x = 3 is a point of a local minimum.
We can do some test on the local maximum and local minimum such as first derivative and
higher order derivative test.


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