Local Maximum and Minimum

Local maximum and minimum are the points of the functions, which give the maximum and minimum range. The local maxima and local minima 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 and local minimum.

Let us learn more about how to find the local maximum and minimum, the methods to find the local maximum and minimum, and the examples of local maximum and minimum.

The local maxima and minima are the input values for which the function gives the maximum and minimum output values respectively. The function equation or the graphs are not sufficiently useful to find the local maxima and local minima points. The derivative of the function is very helpful in finding the local maximum and local minimum of the function.

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, and the input value of \(x_1\), for which \(f(x_2)\) < 0, is called the local minima, and \(f(x_2)\) is the local minimum value. The local maximum and the minimum are calculated for only the defined interval and do not apply to the entire range of the function.

The local maximum and 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 maximum and 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 maximum value or 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 we have the following conditions to identify the local maximum and minimum from the first derivative test.

• 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 maxima.
• 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 minima.
• If f ′(x) does not change significantly as x increases through c, then c is neither a point of local maxima nor a point of local minima. In fact, such a point is called a point of inflection.

The following steps are helpful to complete the first derivative test and to find the limiting points.

• 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.
• 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 minima.

Second Derivative Test

The second derivative test is a systematic method of finding the absolute maximum and absolute minimum value 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 we have the following conditions to identify the local maximum and minimum from the second derivative test.

• x = k, is a point of local maxima if f'(k) = 0, and f''(k) < 0. The point at x= k is the locl maxima and f(k) is called the local maximum value of f(x).
• x = k is a point of local minima if f'(k) = 0, and f''(k) >0 . The point at x = k is the local minima and f(k) is called the local minimum value of f(x).
• The test fails if f'(k) = 0, and f''(k) = 0. And the point x = k is called the point of inflection.

The following sequence of steps facilitates the second derivative test, to find the local maxima and local minima 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 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 minima.
• If the second derivative is lesser than zero \(f''(x_2)<0\), then the limiting point \((x_2)\) is the local maxima.

The following important terms are helpful for a better understanding of local maximum and minimum.

• Maximum: The maximum input value of x, at which the function f(x) has the maximum output, is called the maximum of the function. It is generally defined within an interval and is also called the local maximum.
• Absolute Maximum: The absolute maximum is a point x across the entire range of the function f(x) at which it has a maximum value. The absolute maximum is also sometimes referred to as a global maximum.
• Minimum: The minimum input value of x, at which the function f(x) has the minimum output, is called the minimum of the function. It is generally defined within an interval and is also called the local minimum.
• Absolute Minimum: The absolute minimum is a point x across the entire range of the function f(x) at which it has a minimum value. The absolute minimum is also sometimes referred to as a global minimum.
• Point of Inversion: The value of x within the domain of f(x), which is neither a local maximum nor a local minimum, is called the point of inversion. The points in the immediate neighborhood towards the left and towards the right of the point of inversion, have a slope of zero.
• Maximum Value: The output obtained from the function f(x), on substituting the local maxima point value for x, is called the maximum value of the function. It is the maximum value of the function across the range of the function.
• Minimum Value: The output obtained from the function f(x), on substituting the local minima point value for x, is called the minimum value of the function. It is the minimum value of the function across the range of the function.
• Extreme Value Theorem: For a function f defined in a closed interval [a, b], and is continuous over this closed interval, there exist points c, d, within the interval [a, b] at which this function f attains a maximum and minimum value. f(c) > f(x) > f(d).

Related Topics

The following topics help for a better understanding of local maximum and minimum.

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

How Do You Find the Local Minimum and Maximum?

The local minimum and maximum can be 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 and minimum. The first derivative test or the second derivative test is helpful to find the local minimum and maximum.

What Is the Difference Between Local Maxima and Absolute Maxima?

The local maximum is a point within an interval at which the function has a maximum value. The absolute maxima is also called the global maxima and is the point across the entire domain of the given function, which gives the maximum value of the function.

What Are the Uses of Local Maximum and Minimum?

The local maximum and minimum can be used to find the optimal solution for a real-life problem situation, expressed in the form of an equation. The input values for which these expressions have a maximum or minimum output can be computed from the local maximum and minimum.