Relative Maxima - Definition, Formula, Application, Examples

Relative Maxima

Relative maxima is a point in the domain of the functions, which has the maximum range. The relative maxima can be easily identified from the graph or 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 relative maxima of the function.

Let us learn more about how to find the relative maxima, the methods to find the relative maxima, and the examples of relative maxima.

1.	What Is Relative Maxima?
2.	How To Find Relative Maxima?
3.	Applications of Relative Maxima
4.	Examples of Relative Maxima
5.	Practice Questions on Relative Maxima
6.	FAQs on Relative Maxima

What Is Relative Maxima?

Relative maxima can be easily identified from the graph of the function. It is the turning point in the graph of the function. Relative maxima is a point at which the graph of the function changes direction from increasing to decreasing. Relative Maxims is a point that is higher than the points towards its left, and towards its right.

In the above graph the values of the function at the points c, \(x_1\), and \(x_2\), are f(c), \(f(x_1)\), and \(f(x_2)\) respectively. Here the point 'c' is referred as relative maxima, since its value f(c), is maximumum, in comparison with the other point values of the function.

The relative maxima is the input value for which the function gives the maximum output values. The function equation or the graph of the function is useful to find the relative maxima. Also, the derivative of the function is very helpful in finding the relative maxima of the function. The above graph shows the relative maxima with reference to the other domain points of the function. In addition to this, the function also has a local maximum and a global maximum.

How To Find Relative Maxima?

The relative maxima can easily be found from the graph of the function, by observing the values of the neighboring points, of the relative maxima point. Further the relative maxima 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. 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). Let the function f(x) be continuous at a critical point c in the domain of the function. Here if f ′(x) changes sign from positive to negative as x increases through c. Also let us consider two points \(x_1\), \(x_2\) in the neighborhood of c. If the function values of these neighboring points are lesser than the function value at the point c, then c is the relative maxima.

The following steps are helpful to complete the first derivative test and to find the relative maxima.

• 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 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 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 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 minimum.
• If the second derivative is lesser than zero \(f''(x_2)<0\), then the limiting point \((x_2)\) is the local maximum.

Applications of Relative Maxima

The concept of relative maxima has numerous uses in business, economics, engineering. Let us find some of the important applications of the relative maxima.

• 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.
• 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 can be found using.
• 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 maxima.

Related Topics

The following topics help for a better understanding of relative maxima.

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