Extremum

In calculus, when we have to analyze the behaviour of a function f on an interval I. Does function f have a maximum value? or Does it have a minimum value? Here comes the concept of extremum. An extremum meaning extremum of a function is the point where we find the maximum or minimum value of the function in some interval. It is practically very helpful as it helps us in solving the complex problems of science, engineering, and commerce. Here we will learn the extremum meaning, extremum definition, and a few solved examples of extremum.

Extremum meaning is to find the maximum or minimum value. If a function f (x) is defined on x, then based on the x - interval, the function attains an extremum known as “global” or “local” extremum. 

Extremum of Functions

Extremum of functions refers to the least and the greatest values of the function. There are three different cases for all such values that are discussed below:

Case (i): Consider a function y = f(x) is the strictly increasing function in an interval (a,b). So, f(a) is the least value and f(b) is the greatest value as shown in figure-1.

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Case (ii): Consider a function y = f(x) is the strictly decreasing function in an interval (a,b). So, f(a) is the greatest value and f(b) is the lowest value as shown in figure-2.

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Case (iii): Consider a function y = f(x) is non monotonic in interval [a,b] and is continuous . So  the greatest and the least value of the function are at those points where dy/dx = 0 or where dy/dx does not exist or at extreme values i.e. at x = a and x = b.(figure-3)

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Points to Remember

• A function may consist of many local maxima and local minima but it has only one global maximum and one global minimum.

• The value of the local maximum or local minimum may or may not be the global maxima or global minima.

• The value of the local maximum can be less than a local minimum at some point.

• For any given continuous function, the point of minima and maxima is alternate.

Relative Extrema

A relative maximum point of a function is a point (x,y) on the graph of the function whose y coordinate is larger than all other y coordinates on the graph at points “close to (x,y). (x,f(x)) is relatively maximal if there is an interval (a,b) with a<x<b and \[f(x)\leq f(z)\]for every z in (a,b). A relative extremum is either a relative minimum or a relative maximum.

• The plural of extremum is extrema and similarly for maximum and minimum.

• As we know relative extremum is “extreme” locally by looking at points “close to” it, it is also referred to as a local extremum.

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Relative maximum and minimum points are quite distinctive on the graph of a function and are therefore useful in understanding the shape of the graph. In many applied problems we want to find the largest or smallest value that a function achieves (for example, we might want to find the minimum cost at which some task can be performed) and so identifying maximum and minimum points will be useful for applied problems as well. 

What is a Critical Point?

The critical point of a function corresponds to any value in its domain where its derivative value is 0. We can say that every local extremum is a critical point but every critical point need not be a local extremum. So, if we have a function that is continuous, it must have maxima and minima or local extrema. So we can say that every such function will have critical points. In case if the given function is monotonic, the maximum and minimum values lie at the endpoints of the domain of the definition of that particular function.

Following are the Guideline that needs to follow for Finding Absolute Extrema

Given Continuity of function f  and Closed Interval.

Step 1: Verify the function is continuous on (a,b)

Step 2: Find the derivative and then determine all critical values of f that are in (a.b)

Step 3:  Evaluate the function at the critical values found in Step 2 and the endpoints x=a and x=b of the interval.

Step 4: The absolute maximum and minimum value of function f correspond to the largest and smallest y-values respectively that are found in Step 3.

Solved Examples:

1.Find the extremum of function \[f(x) = 3x^{3} - 9x\] in the interval [-1,4]

Ans: Given function \[f(x) = 3x^{3} - 9x\] 

To find the extremum value of the function we have to differentiate the given function and equate the differentiated value to 0

\[= f(x) = 3x^{3} - 9x\] 

\[\Rightarrow f`(x) = 3 \times 3x^{2} - 9\]

Taking 9 common from both the terms

\[\Rightarrow f`(x) = 9(x^{2} - 1)\]

Now\[f`(x) = 0\]

\[\Rightarrow 9(x^{2} - 1) = 0\]

\[\Rightarrow x = \pm 1\]

Now we will find value at \[\pm 1\]

f(-1)= \[3x^{3} - 9x\]

        =3\[(-1)^{3} - 9 \times (-1)\]

        =-3+9

        =-6

f(1) = \[3x^{3} - 9x\]

      =\[3 \times (1)^{3} - 9 \times 1\]

      =3-9

      =-6

Now we will find value at 4

f(4)=\[3x^{3} - 9x\]

      =\[3 \times (4)^{3} - 9 \times 4\]

      =\[3 \times 64 - 36\]

      =156

Hence, Greatest value = 156 and least value = – 6.

2.Find the extremum value of function \[f(x) = x^{3} - 6x^{2} + 9x + 15\]

Sol: Given function \[f(x) = x^{3} - 6x^{2} + 9x + 15\]

To find extremum value differentiate the given function and equate it to 0

Differentiating with respect to x

\[f`(x) = 3x^{2} - 12x + 9\]

Taking 3 common we have

\[f`(x) = 3(x^{2} - 4x + 3)\]

\[f`(x) = 3(x - 3)(x - 1)\]

Now equate f’(x) value to 0

\[f`(x) = 0\]

\[\Rightarrow 3(x - 3)(x - 1) = 0\]

\[\Rightarrow x = 3,1\]

At x = 1, f’(x) changes from positive to negative

Since, x = 1 is a point of Maxima

At x = 3, f‘(x) changes from negative to positive

Since, x = 3 is the point of Minima.

Local maxima value\[f(1) = (1)^{3} - 6(1)^{2} + 9(1) + 15 = 19\]

Local minima value \[f(3) = (3)^{3} - 6(3)^{2} + 9(3) + 15 = 27 - 54 + 27 + 15 = 15\]

Hence the minimum value is 15 and maximum value is 19.

Conclusion

The maxima and minima are collectively called “Extrema”. Maxima and minima are very important concepts in the calculus of variations, as it helps to find the extreme values of a function. The theory behind finding the maximum and minimum values of a function is based on the derivative of a function that is equal to the slope of the tangent.