Finding The Local Minimum - Cuemath

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 real-valued 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 real-valued 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.

• Derivative Formula
• Differentiation
• Mean Value Theorem
• Rolle's Theorem
• Differential Equations Formula
• Application of Derivatives