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Relative Minima
Relative minima is a point in the domain of the functions, which has the minimum value. The relative minima 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 minima of the function.
Let us learn more about how to find the relative minima, the methods to find the relative minima, and the examples of relative minima.
1.  What Is Relative Minima? 
2.  How To Find Relative Minima? 
3.  Applications of Relative Minima 
4.  Examples of Relative Minima 
5.  Practice Questions on Relative Minima 
6.  FAQs on Relative Minima 
What Is Relative Minima?
Relative minima can be easily identified from the graph of the function. It is the turning point in the graph of the function. Relative minima is a point at which the graph of the function changes direction from decreasing to increasing. Relative minim is a point that is lower than the points towards its left, and towards its right.
From the 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 minima, since its value f(c), is minimum, in comparison with the other point values of the function.
The relative minima is the input value for which the function gives the minimum output values. The function equation or the graph of the function is useful to find the relative minima. Also, the derivative of the function is very helpful in finding the relative minima of the function. The above graph shows the relative minima with reference to the other domain points of the function. In addition to this, the function also has a local minimum and a global minimum.
How To Find Relative Minima?
The relative minima can easily be found from the graph of the function, by observing the values of the neighboring points, of the relative minima point. Further, the relative 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 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 minimum 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 negative to positive 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 greater than the function value at point c, then c is the relative minima.
The following steps are helpful to complete the first derivative test and to find the relative minima.
 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 relative minima.
Second Derivative Test
The second derivative test is a systematic method of finding the relative minima 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 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 the function f(x).
The following sequence of steps facilitates the second derivative test, to find the relative 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 find 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 Minima
The concept of relative minima has numerous uses in business, economics, engineering. Let us find some of the important applications of relative minima.
 The minimum number of seeds to be sown in a field to get the optimum yield can be found from the relative minima.
 From a parabolic equation, the relative minima help in knowing the point at which the vertex of the parabola lies.
 The lowest point reached by a swing can be found with the help of the relative minima.
 The price of a stock, if represented in the form of a functional equation, or a graph can be used to find the relative minima, which is the lowest price at which the stock can be purchased.
 The lowest voltage at which an electrical appliance does not function can be identified by finding the relative minima of the voltage function.
 In the food processing units, the minimum humidity required to keep the food fresh can be found from the relative minima of the humidity function.
Related Topics
The following topics help for a better understanding of relative minima.
Examples on Relative Minima

Example 1: Find the relative minima 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 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, compared to its neighboring points..
Therefore, the relative minima of the function is at x = 1.

Example 2: Find the relative minima 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 relative maxima.
f''(3) = 6(3)  12 = 18  12 = 6, f''(3) < 0, and x = 3 is the relative minima.
Therefore by using the second derivative test, the relative minima is at x = 3.
FAQs on Relative Minima
How Do You Find Relative Minima?
Relative minima can be easily identified from the graph of the function. It is the turning point in the graph of the function. Relative minima is a point at which the graph of the function changes direction from decreasing to increasing. Relative minima is a point that has lower function values than the points in its neighborhood.
Is Local Minimum the Same as Relative Minima?
The local minimum is different from relative minima. The local minimum is across a defined interval, and the relative minima is with reference to its neighboring points. Both the local minimum and the relative minima are turning points with reference to the graph of the function, and in rare cases, the local minimum and the relative minima are the same values. The local minimum can be founding using the first derivative test, or the second derivative test. The relative minima can be easily identified from the graph of the function.
What Are the Methods To Find Relative Minima?
The relative minima can be easily identified from the graph of the function. Further, the relative minima can also be computed using the first derivative test, or the second derivative test.
What Is the Use of Relative Minima?
The relative minima helps in finding a point in the domain of a function, which has an optimal output. Relative minima has numerous applications in business, economics, physics, engineering. The following are a few quick examples of relative minima.
 The price of a stock, if represented in the form of a functional equation or a graph, the relative minima is helpful to find the point where the price of the stock is minimum.
 The low current flow based on the voltage in an electrical appliance, at which it is the lowest, can be identified with the help of the relative minima, of the voltage function.
 In the food processing units, the humidity is represented by a function, and the minimum humidity at which the food remains fresh, can be found using the relative minima .
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