Local Minimum
Local minimum is the point in the domain of the functions, which has the minimum value. The local minimum 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 minimum for a function.
Let us learn more about how to find the local minimum, the methods to find local minimum, and the examples of local minimum.
1.  What Is Local Minimum? 
2.  Methods to Find Local Minimum 
3.  Uses of Local Minimum 
4.  Examples of Local Minimum 
5.  Practice Questions on Local Minimum 
6.  FAQs on Local Minimum 
What Is Local Minimum?
The local minimum is the input value for which the function gives the minimum output values. The function equation or the graph of the function is not sometimes sufficient to find the local minimum. The derivative of the function is very helpful in finding the local minimum of the function. The below graph shows the local minimum within the defined interval of the domain. Further, the function has another minimum value across the entire range, which is called the global minimum.
Let us consider a function f(x). The input value of \(x_1\) for which \(f(x_1)\) < 0, is called the local minimum, and \(f(x_1)\) is the local minimum value . The local minimum is calculated for only the defined interval and does not apply to the entire range of the function.
Methods to Find Local Minimum
The local minimum 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 minimum. 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 minimum 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 negative to positive 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 minimum.
The following steps are helpful to complete the first derivative test and to find the local minimum.
 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 negative for the neighboring point to the left, and it is positive for the neighboring point to the right, then the limiting point is the local minimum.
Second Derivative Test
The second derivative test is a systematic method of finding the local minimum 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 minimum, if f'(k) = 0, and f''(k) > 0. The point at x= k is the local minimum, and f(k) is called the local minimum value of the function f(x).
The following sequence of steps facilitates the second derivative test, to find the local minimum 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 get 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 Minimum
The concept of local minimum has numerous uses in business, economics, engineering. Let us find some of the important uses of the local minimum.
 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 falls, or is minimum.
 The drop in voltage in an electrical appliance, at which it may the functioning of the equipment, can be identified from the local minimum.
 In the food processing units, the minimum humidity to be maintained to keep the food fresh, can be found from the local minimum of the graph of the humidity function.
 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 minimum.
 For a parabolic equation, the local minimum helps in knowing the point at which the vertex of the parabola lies.
 The minimum temperature to be maintained in the fridge can be found from the local minimum of the temperature function.
Related Topics
The following topics help for a better understanding of the local maximum.
Examples on Local Minimum

Example 1: Using the first derivative test, find the local minimum 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 = 1. The points are {0, 2 }.
f'(0) = 6((0)^{2} +(0)  2) = 6(2) = 12, and f'(2) = 6(2^{2} + 2 2) = 6(4) = +24
The derivative of the function is negative towards the left of x = 1, and is positive towards the right. Hence x = 1 is the local minimum.
Therefore, the local minimum is at x = 1.

Example 2: What is the local minimum and the minimum value of the function f(x) = x^{3}  6x^{2}+9x + 15? Here use the second derivative test, to find the local minimum.
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.
f''(3) = 6(3  2) = 6(1) = 6, f''(3) > 0, and x = 3 is the minima
The minimum value of the function is f(3) = 3^{3}  6(3)^{2} + 9(3) + 15 = 27  54 + 27 + 15 = +15
Therefore by using the second derivative test, the local minimum is 3, and the minimum value is f(3) = 15.
FAQs on Local Minimum
How Do You Find The Local Minimum?
The local minimum is found by differentiating the function and finding the turning points at which the slope is zero. The local minimum is a point in the domain, which has the minimum value of the function. The first derivative test or the second derivative test is helpful to find the local minimum of the given function.
What Is the Difference Between Local Minimum and Relative Minima?
The local minimum is a point within an interval at which the function has a minimum value. The relative minima is the minimum point in the domain of the function, with reference to the points in the immediate neighborhood of the given point.
What Are the Methods To Find Local Minimum?
The two important methods to find the local minimum are the first derivative test and the second derivative test. The first derivative test is the approximate method to find the local minimum, and the second derivative test is a systematic process of finding the local minimum.
What Is the Use of Local Minimum?
The local minimum is used to find the optimal value of a function. The concept of local minimum is used in business, economics, physical, and engineering. A local minimum is used to find the lowest price at which a stock can be bought, to find the minimum voltage required for an electrical appliance, or to find the minimum storage temperature of food commodities.
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