Extremum Definition
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.
FAQs on Extremum
1. What is an extremum of a function in calculus?
In calculus, an extremum refers to a point where a function reaches its maximum or minimum value. These points are crucial for understanding the behaviour of a function. An extremum can be a local (or relative) extremum, which is the highest or lowest point in a specific neighbourhood of the function, or a global (or absolute) extremum, which is the overall highest or lowest point across the function's entire domain.
2. What is the difference between a local extremum and a global extremum?
A local extremum is a point on a function's graph that is a maximum or minimum within a small, immediate interval around it, much like the highest peak in a single mountain range. In contrast, a global extremum is the single highest or lowest value the function attains over its entire defined domain, similar to the highest point on an entire continent. A function can have several local extrema, but it can only have one global maximum and one global minimum.
3. What are the general steps to find the extremum of a function on a closed interval?
To find the absolute extremum of a continuous function f(x) on a closed interval [a, b], you can follow these steps as per the CBSE/NCERT curriculum for the 2025-26 session:
- Step 1: Find the first derivative of the function, f'(x).
- Step 2: Identify all critical points. These are the points within the interval (a, b) where the derivative f'(x) is equal to zero or is undefined.
- Step 3: Evaluate the function f(x) at all the critical points found in the previous step.
- Step 4: Evaluate the function f(x) at the endpoints of the interval, i.e., at x=a and x=b.
- Step 5: Compare all the values calculated in steps 3 and 4. The largest value is the absolute maximum (global extremum), and the smallest value is the absolute minimum.
4. Why is every local extremum a critical point, but not every critical point is a local extremum?
This is a key concept in calculus. A local extremum (a peak or a valley) must occur where the function's slope either becomes zero (a horizontal tangent) or is undefined (a sharp corner). Both these conditions define a critical point. Therefore, every local extremum is a critical point. However, the reverse is not always true. A function can have a critical point that is neither a maximum nor a minimum. For example, the function f(x) = x³ has a critical point at x=0 because f'(0)=0, but this point is an inflection point where the curve's concavity changes, not an extremum.
5. How is the concept of an extremum applied in real-world problems?
The importance of finding an extremum lies in its wide range of practical applications for optimisation. For example:
- In Business and Economics, it is used to find the production level that yields maximum profit or minimum cost.
- In Engineering, it helps determine the maximum load a structure can bear or the minimum material required to build an object.
- In Physics, it can be used to calculate the maximum height reached by a projectile or the minimum potential energy of a system.
- In Medicine, it can help model the maximum concentration of a drug in the bloodstream over time.
6. What is the difference between the First and Second Derivative Tests for identifying an extremum?
Both tests are used to classify a critical point, but they work differently. The First Derivative Test examines the sign of the derivative f'(x) on either side of a critical point. If the sign changes from positive to negative, it's a local maximum. If it changes from negative to positive, it's a local minimum. The Second Derivative Test uses the function's concavity at the critical point. If the second derivative f''(x) is negative at that point, it indicates a local maximum (concave down). If f''(x) is positive, it indicates a local minimum (concave up). The second test is often faster but can be inconclusive if f''(x) is zero.
7. What is the difference between the terms 'extremum' and 'extrema'?
The distinction is simply a matter of grammar. Extremum is the singular form and refers to a single maximum or minimum value. For example, 'The function has an extremum at x=2.' On the other hand, extrema is the plural form and is used to refer to all the maximum and minimum values of a function collectively. For example, 'We need to find all the extrema of the function on the given interval.'
