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Local Maximum
Local maximum is the point in the domain of the functions, which has the maximum range. The local maximum can be computed by finding the derivative of the function. The first derivative test, and the second derivative test, are the two important methods of finding the local maximum for a function.
Let us learn more about how to find the local maximum, the methods to find local maximum, and the examples of local maximum.
1.  What Is Local Maximum? 
2.  Methods to Find Local Maximum 
3.  Uses of Local Maximum 
4.  Examples of Local Maximum 
5.  Practice Questions 
6.  FAQs on Local Maximum 
What Is Local Maximum?
The local maxima is the input value for which the function gives the maximum output values. The function equation or the graph of the function is not sufficient to find the local maximum. The derivative of the function is very helpful in finding the local maximum of the function. The below graph shows the local maximum within the defined interval of the domain. Further, the function has another maximum range value across the entire domain, which is called the global maximum.
Let us consider a function f(x). The input value of \(x_1\) for which \(f(x_1)\) > 0, is called the local maxima, and \(f(x_1)\) is the local maximum value . The local maximum is calculated for only the defined interval and do not apply to the entire range of the function.
Methods to Find Local Maximum
The local maximum can be identified by taking the derivative of the given function. The first derivative test and the second derivative test are useful to find the local maximum. Let us understand more details, of each of these tests.
First Derivative Test
The first derivative test helps in finding the turning points, where the function output has a maximum value. For the first derivative test. we define a function f(x) on an open interval I. Let the function f(x) be continuous at a critical point c in the interval I. Here if f ′(x) changes sign from positive to negative as x increases through c, i.e., if f ′(x) > 0 at every point sufficiently close to and to the left of c, and f ′(x) < 0 at every point sufficiently close to and to the right of c, then c is a point of local maximum.
The following steps are helpful to complete the first derivative test and to find the local maximum.
 Find the first derivative of the given function, and find the limiting points by equalizing the first derivative expression to zero.
 Find one point each in the neighboring left side and the neighboring right side of the limiting point, and substitute these neighboring points in the first derivative functions.
 If the derivative of the function is positive for the neighboring point to the left, and it is negative for the neighboring point to the right, then the limiting point is the local maxima.
Second Derivative Test
The second derivative test is a systematic method of finding the local maximum of a realvalued function defined on a closed or bounded interval. Here we consider a function f(x) which is differentiable twice and defined on a closed interval I, and a point x= k which belongs to this closed interval (I). Here x = k, is a point of local maximum, if f'(k) = 0, and f''(k) < 0. The point at x= k is the local maximum, and f(k) is called the local maximum value of the function f(x).
The following sequence of steps facilitates the second derivative test, to find the local maxima and local minima of the realvalued function.
 Find the first derivative f'(x) of the function f(x) and equalize the first derivative to zero f'(x) = 0, to the limiting points \(x_1, x_2\).
 Find the second derivative of the function f''(x), and substitute the limiting points in the second derivative\(f''(x_1), f''(x_2)\)..
 If the second derivative is greater than zero\(f''(x_1) > 0\), then the limiting point \((x_1)\) is the local minimum.
 If the second derivative is lesser than zero \(f''(x_2)<0\), then the limiting point \((x_2)\) is the local maximum.
Uses of Local Maximum
The concept of local maximum has numerous uses in business, economics, engineering. Let us find some of the important uses of the local maximum.
 The price of a stock, if represented in the form of a functional equation and a graph, is helpful to find the points where the price of the stock is maximum.
 The voltage in an electrical appliance, at which it peaks can be identified with the help of the local maximum, of the voltage function.
 In the food processing units, the humidity is represented by a function, and the maximum humidity at which the food is spoilt can be found using the local maximum.
 The number of seeds to be sown in a field to get the maximum yield can be found with the help of the concept of the local maximum.
 For a parabolic equation, the local maximum helps in knowing the point at which the vertex of the parabola lies.
 The maximum height reached by a ball, which has been thrown in the air and following a parabolic path, can be found by knowing the local maximum.
Related Topics
The following topics help for a better understanding of local maximum.
Examples on Local Maximum

Example 1: Using the first derivative test, find the local maximum of the function y = 2x^{3} + 3x^{2}  12x + 5.
Solution:
The given function is y = f(x) = 2x^{3} + 3x^{2}  12x + 5
f'(x) = 6x^{2} + 6x  12
f'(x) = 0; 6x^{2}  6x  12 = 0, 6(x^{2} + x  2) = 0, 6(x  1)(x + 2) = 0
Hence the limiting points are x = 2, and x = 1.
Let us now take the points in the immediate neighborhood of x = 2. The points are {3, 1}.
f'(3) = 6((3)^{2} + (3)  2) = 6(4) = +24, and f'(1) = 6((1)^{2} + (1) 2) = 6(2) = 12
The derivative of the function is positive towards the left of x = 2, and is negative towards the right. Hence x = 2 is the local maxima.
Therefore, the local maximum is 2.

Example 2: What is the local maximum and the maximum value of the function f(x) = x^{3}  6x^{2}+9x + 15? Here use the second derivative test, to find the local maximum.
Solution:
The given function is f(x) = x^{3}  6x^{2}+9x + 15.
f'(x) = 3x^{2}  12x + 9
f'(0) = 3(x^{2}  4x + 3)
x^{2}  4x + 3 = 0 or (x  1)(x  3)=0.
Here x = 1, and x = 3
Here using the second derivative test we have f''(x) = 6x  12
f''(1) = 6(1)  12 = 6  12 = 6., f''(1) < 0, and x = 1 is the maxima.
Therefore by using the second derivative test, the local maximum is 1, and the maximum value is f(1) = 19.
FAQs on Local Maximum
How Do You Find The Local Maximum?
The local maximum is found by differentiating the function and finding the turning points at which the slope is zero. Further, these turning points can be checked through different methods to find the local maximum. The first derivative test or the second derivative test is helpful to find the local maximum of the given function.
What Is the Difference Between Local Maximum and Relative Maxima?
The local maximum is a point within an interval at which the function has a maximum value. The relative maxima is the maximum point in the domain of the function, with reference to the points in the immediate neighborhood of the given points.
What Are the Methods To Find Local Maximum?
The two important methods to find the local maximum is the first derivative test, and the second derivative test. The first derivative test is the approximate method to find the local maxima, and the second derivative test is a systematic process of finding the local maximum.
What Is the Use of Local Maximum?
The local maximum is used to find the optimal value of a function. The concept of local maximum is used in business, economics, physical and engineering. Local maximum is used to find the optimal price of a stock, to find the peak break down voltage of an electrical appliance, or to find the optimal storage temperature of food commodities.
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