Relative Maxima and Minima

Relative maxima and minima are the points of the functions, which give the maximum and minimum range. The relative maxima and minima is computed with reference to the other points in its neighborhood. It 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 relative maxima and minima, the methods to find maxima and minima, and the examples on relative maxima and minima.

The relative 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 relative maxima and minima points. The derivative of the function is very helpful in finding the relative maxima and relative minima of the function.

Let us consider a function f(x). The input value of \(c_1\) for which \(f(c_1)\) > \(f(x_1)\) and \(f(x_2)\), with reference to the neighboring points \(x_1\), and \(x_2\) , is called the relative maxima, and \(f(c_1)\) is the maximum value. Also for the input value of \(c_2\), for which \(f(c_2)\) < \(f(x_3)\) and \(f(x_4)\), with reference to the neighboring points \(x_3\), and \(x_4\),  is called the relative minima, and \(f(c_2)\) is the minimum value. The relative maxima and minima are calculated for only with reference to the neighboring points and do not apply to the entire range of the function.

The relative maxima and minima 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 relative maxima and minima. 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 relative maxima and minima 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 a point sufficiently close to and to the left of c, and f ′(x) < 0 at a point sufficiently close to and to the right of c, then c is a point of relative maxima.
• If f ′(x) changes sign from negative to positive as x increases through c, i.e., if f ′(x) < 0 at a point sufficiently close to and to the left of c, and f ′(x) > 0 at a point sufficiently close to and to the right of c, then c is a point of relative 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 relative 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 relative minima.

Second Derivative Test

The second derivative test is a systematic method of finding the relative maxima and minima 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 relative maxima and minima from the second derivative test.

• x = k, is a point of relative maxima if f'(k) = 0, and f''(k) < 0. The point at x= k is the relative maxima and f(k) is called the maximum value of f(x).
• x = k is a point of relative minima if f'(k) = 0, and f''(k) >0 . The point at x = k is the relative minima and f(k) is called the 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 relative maxima and 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 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 relative minima.
• If the second derivative is lesser than zero \(f''(x_2)<0\), then the limiting point \((x_2)\) is the relative maxima.

The following important terms are helpful for a better understanding of relative maxima and minima.

• Local 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.
• Local 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 maximum 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 minimum 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).

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 and minimum.
• The voltage in an electrical appliance, at which it peaks can be identified with the help of relative maxima, 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, and the minimum humidity required to keep the food fresh, can be found from the relative maxima and minima.
• The number of seeds to be sown in a field to get the maximum output can be found from the relative maxima.
• For a parabolic equation, the relative maxima and minima is helpful 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 relative maxima.

Related Topics

The following topics help in a better understanding of relative maxima and minima.

• Local Maximum and Minimum
• Local Maximum
• Relative Maxima
• First Derivative Test
• Second Derivative Test
• Application of Derivatives

How Do You Find the Relative Minima and Maxima?

The relative maxima and minima 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 relative maxima and minima. The first derivative test or the second derivative test is helpful to find the relative maxima and minima.

What Is the Difference Between Relative Maxima and Absolute Maxima?

The relative maxima is a point across a set of points, 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 Relative Maxima and Minima?

The relative maxima can be used to find the optimal solution for a real-life problem situation, expressed in the form of an equation. The price of a stock, the humidity levels for food storage, the breakdown voltage for electric equipment, can easily be calculated with the help of the relative maxima and minima of the respective functions.