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 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 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 realvalued 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.
Examples on Relative Maxima

Example 1: Find the relative maxima of the function f(x) = 2x^{3} + 9x^{2}  24x + 30.
Solution:
The given function is f(x) = 2x^{3} + 9x^{2}  24x + 30.
Let us take the first derivate of this function, to find the relative maxima of the function.
f'(x) = 6x^{2} + 18x 24. Here we equalize this derivative to zero, to find the limiting points.
f'(x) = 0, or 6x^{2} + 18x  24 = 0
6(x^{2} + 3x  4 = 0)
6(x  1)(x + 4) = 0
x = 1, and x = 4. are the limiting points.
The points in the neighbourhood of x = 1, are {0, 2}.
The values of the function for these points are f(0) = 30, f(1) = 17, f(2) = 34.
Here the limiting point x = 1, has the least value.
The points in the neighbourhood of x = 4 are {1, 3}
The values of the function for these points are f(4) = +142, f(5) = +130, f(3) = +129.
Here x = 4 has the maximum value and it is the relative maxima.
Therefore, the relative maxima of the function is 4.

Example 2: Find the relative maxima of the function f(x) = x^{3}  6x^{2}+9x + 15, by using the second derivative test.
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.
Therefore by using the second derivative test, the relative maxima is at x = 1.
FAQs on Relative Maxima
How Do You Find 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 Maxima is a point that is having higher than the points in its neighborhood.
Is Local Maximum the Same as Relative Maxima?
The local maximum is different from relative maxima. The local maximum is across a defined interval, and the relative maxima is with reference to the neighboring points. Both the local maximum and the relative maxima are turning points with reference to the graph of the function. The local maximum can be founding using the first derivative test, or the second derivative test. The relative maxima can be easily identified from the graph of the function.
What Are the Methods To Find Relative Maxima?
The relative maxima can be easily identified from the graph of the function. Further, the relative maxima can also be computed using the first derivative test, or the second derivative test.
What Is the Use of Relative Maxima?
The relative maxima helps in finding a point in the domain of a function, which has an optimal output. Relative maxima has numerous applications in business, economics, physics, engineering. The following are a few quick examples of relative maxima.
 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.
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