Introduction to Maxima and Minima
Maxima and Minima are the most important topics in differential calculus. A division of Mathematics known as “Calculus of Variations” tackles the maxima and the minima of the functionals. The calculus of variations is affected by the changes in the functionals, in which minor variation in the function brings about variation in the functional value. The first variation is stated as the linear part of the variation in the functional, and the second part of the variation is stated in the quadratic part. Functionals are often determined as the definite integrals which include both the functions and their derivatives. The functions that maximize or minimize the functionals can be determined through the Euler – Lagrange of the calculus of variations. The two Latin words i.e. maxima and minima usually mean the maximum and minimum value of a function respectively. The maxima and minima are known as “Extrema”. Here, we are assuming that our function will be continuous for its entire domain. Let us first learn first about derivatives before learning how to determine maxima and minima.
Maxima and Minima
Maxima and minima are called the extremes of a function. There are two maximums and two minima for every function within a set of ranges. With respect to the function, its maximum and minimum values are known as the absolute maxima and the absolute minima, respectively.
Another maximum and minimum of a function are known as local maxima and local minima because they are not the absolute maxima and minima of the function. Find the maxima and minima of a function and learn more about local maxima and minima.
The maxima and minima in the curve of a function are the peaks and valleys. It is possible for a function to have as many maxima and minima as it needs. Calculus allows us to calculate the maximum and minimum values of any function without ever having to look at its graph. The maxima of the curve will be the highest point within the given range, and the minima will be the lowest. A combination of maxima and minima is extreme.
In a function, maxima and minima can be divided into two types:
Local Maxima and Minima
Absolute or Global Maxima and Minima
Maxima and minima in a particular interval are called local maxima and minima. In a particular interval, a local maxima would be where values of a function near a particular point are always less than the value of the function at that same point. Local minima, on the other hand, would be the value of the function at a point where the value of the function near that point is greater than its value at that point.
Local Maxima and Local Minima
A local maximum point on a function is a point (x,y) on the graph of the function whose y coordinate is greater than all other y coordinates on the graph at points "close by'' (x,y).
In other way, (x,f (x)) is a local maximum and if there is an interval (a,b) with a < x< b and f(x) ≥ f(z) for every z in (a,b). Similarly, (x,y) will be determined as the local minimum point if it has locally the smallest y coordinate.
To define it more precisely, (x,f(x)) is considered as a local minimum if there is an interval (a,b) with a < x < b and f(x) ≤ f(z) for every z in (a,b). A local extreme is either a local minimum or a local maximum.
Local maximum and minimum points are completely different on the graph of a function, and it is beneficial to understand the shape of the graph. In various problems, we are required to determine the greatest or smallest value that a function attains. For example, we might carry out some tasks to determine the maximum point. Hence, observing maximum and minimum points will also be beneficial for applied problems. Some examples of local maxima and minima are given in the below figure:
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If (x, f(x)) is a point where f(x) approaches a local maximum or minimum, and if the derivative of f is placed at x, then the graph must be having a tangent line and the tangent line which is formed must be horizontal.
Maximum and Minimum Absolute Values
When a function is defined over an entire domain, its greatest point is known as the absolute maximum of the function, while its lowest point is known as the absolute minimum. An absolute maximum and absolute minimum of a function can only occur in an entire domain. Alternatively, the maxima and minima of the function can be called global maxima and global minima.
What are the Maxima and Minima of a Function?
The first- and second-order derivative tests can be used to calculate a function's maximum and minimum values. Maxima and minima of a function can be found using derivative tests. We will go through each of them in turn.
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Maxima and Minima: First Order Derivative Test
Taking the first derivative of a function gives the slope of the function. As we head towards a maximum point, we find that the slope of the curve increases as we get closer to that point, then becomes zero at the maximum point, and then decreases as we move away from it. Similar to the minimum point, as we move towards the minimum point, the slope of the function decreases, becoming 0 at the minimum point, and then increasing as we move away from the minimum point. Based on this information, we can determine whether a point is a maximum or a minimum.
Let us consider a function f, which is continuous at the critical point, defined in an open interval I, and f'(c) = 0 (slope at c = 0). When we check the values of f'(x) at the points left and right of the curve, and check the nature of f'(x), we can infer that the given point will be:
Local Maxima: If f'(x) changes sign from positive to negative as x increases via point c, then f(c) represents the maximum value of the function in that range.
Local Minima: If f’(x) changes sign from negative to positive as x increases via point c, then f(c) gives the minimum value of the function within that range.
Inflection Point: if f'(x) doesn't change with x increasing via c, and point c is neither the maximum nor the minimum of the function, then point c is the inflection point.
Maxima and Minima: Second Order Derivative Test
A second-order derivative test for maxima and minima tests whether the slope is equal to 0 at the critical point x = c (f'(c) = 0), at which point we find the second derivative of the function. Within the given range, if the second derivative of the function is present:
Local maxima: If f''(c) < 0
Local minima: If f''(c) > 0
Test fails: If f''(c) = 0
Some Important Notes on Maxima and Minima
The maxima and minima in a function are the highest and lowest points.
A function can have only one absolute maximum and one absolute minimum over its entire domain.
If a function f is either increasing in I or decreasing in I then it is called a monotonous function in the interval I.
Nature of Derivatives
Let us consider a point M where x = a and now we will make an effort to determine the nature of the derivatives. There are altogether four possibilities:
If the value of f’(a) = 0, then the tangent is drawn parallel to the x−axis i.e. the slope will be zero. There are three possible situations.
The value of f when compared with the value of f at M, increases if moved towards the right or left of M (Local minima: resembles valleys)
The value of f when compared to the value of f at M, decreases if moved towards the right or left of M (Local maxima: resembles hills)
The value of f when compared with the value of f at M, either increases and decreases as moved towards the left and right respectively of M (Neither: resembles a flat land)
If, the tangent is formed at a positive slope. The value of f'(a), when compared to the value of at M, increases if moved towards the right and decreases if moved towards the left. So, in this condition, it is not possible to determine any local extrema.
If, the tangent is formed at a negative slope. The value of f'(a), when compared to the value of f'(a) at M, increases if moved towards the left and decreases if moved towards the right. So, in this condition, it is not possible to determine any local extrema.
f′ does not exist at point M i.e. the function is not differentiable at M. It usually materializes when you find a sharp corner somewhere in the graph of f. All the three scenarios discussed in the earlier points also hold true for this point.
Different Possibilities of Derivative Function Table
Solved Examples
1. Determine the Local Maxima and Minima for the Function y = x³ - 3x + 2.
Solution: We are required to determine the critical points for this function. For which, we will calculate the df/dx as follows:
y = x³ -3x + 2
dy/dx = 3x² - 3
At critical points, dy/dx = 0, we have
3x² - 3 = 0
3(x² - 1) = 0
(x-1)(x+1) = 0
x = 1 , x = -1
Now, we will find whether any of these stationary points are extreme points. We will apply a second derivative test for this.
dy/dx = 3x² - 3
d²y/d²x = 6x
For x = 1 ; dy/dx = 6/times 1 = 6. Hence, the point (1,y(x = 1) is a point of local maxima.
For x = -1 ; dy/dx = 6/times -1 = -6. Hence, the point (-1,y(x = -1) is a point of local maxima.
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2. Determine the Local Maxima and Local Minima for all the Functions f(x) = x³- x.
Solution: The derivative of the function f’(x) is -3x-1 .It is defined everywhere and value is zero at x = 3–√3/3
By initially looking at x = 3–√3/3, we can see that f(3–√3/3) = -2
3–√3/9. Now, we will check two points placed at either side of x = 3–√3/3 by ensuring that no value is far away from the critical value. As,
3–√3< 3 and
3–√3/3 < 1 and we can make use of x = 0 and x=1. As f(0) = 0 > -2
3/9−−−√3/9 and f(1) = 0 > -2
3/9−−−√3/9, there should be a local minimum at x = (3–√3/3 ). For x = (3–√3/3 ), we can see that f - (3–√3/3 ) = 2
3 – √ 3 /9. For this, we will use x = 0 and x= -1and we will determine f(-1) = f(0) = 0 < 2 3 – √ 3 /9, so there should be local maxima value at x = - ( 3 – √ 3 /3 )
Quiz Time
1. Identify the relative maximum point in the below graph.
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(0,3)
(3,0)
(1,4)
(4,1)
2. At which coordinates, function is decreasing in the below graph?
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(4, ∞ )
(-4, ∞)
(-∞, 4)
4 < x < 6
FAQs on Local Maxima and Minima
1. What is meant by local maxima and minima in calculus?
In calculus, a local maximum is a point on a function's graph that is higher than all other nearby points in its immediate vicinity or neighborhood. Conversely, a local minimum is a point that is lower than all other nearby points. These points resemble the peaks and valleys of a curve, respectively. They are not necessarily the absolute highest or lowest points over the entire function, but only within a specific, localized interval.
2. What is the primary difference between a local extremum and an absolute extremum?
The primary difference lies in the scope of comparison.
- A local extremum (maximum or minimum) is the highest or lowest point within a small, open interval or neighborhood around it. A function can have multiple local maxima and minima.
- An absolute extremum is the highest or lowest point over the function's entire defined domain. A function can have only one absolute maximum and one absolute minimum.
3. How do we find the local maxima and minima of a function using the First Derivative Test?
The First Derivative Test helps identify local extrema by observing the change in the sign of the first derivative (f'(x)) around a critical point 'c' (where f'(c) = 0 or is undefined):
- Local Maximum: If f'(x) changes sign from positive to negative as x passes through 'c', then the function has a local maximum at that point. This means the function was increasing and then started decreasing.
- Local Minimum: If f'(x) changes sign from negative to positive as x passes through 'c', then the function has a local minimum. This indicates the function was decreasing and then started increasing.
4. What is the Second Derivative Test and when is it used?
The Second Derivative Test is often a quicker method to classify critical points as local maxima or minima, provided the second derivative exists and is not zero. For a critical point 'c' where f'(c) = 0:
- If f''(c) < 0 (negative), the function is concave down at that point, indicating a local maximum.
- If f''(c) > 0 (positive), the function is concave up at that point, indicating a local minimum.
- If f''(c) = 0, the test is inconclusive, and you must use the First Derivative Test to determine the nature of the point.
5. Why does the sign of the first derivative (f'(x)) change around a point of local maxima or minima?
The first derivative, f'(x), represents the slope of the tangent to the curve. For a function to form a peak (local maximum), its slope must transition from positive (increasing function) to zero (at the peak) and then to negative (decreasing function). This change from positive to negative slope is why the sign of f'(x) changes. Similarly, for a valley (local minimum), the slope must transition from negative (decreasing) to zero (at the bottom) and then to positive (increasing), causing the sign of f'(x) to change from negative to positive.
6. What happens if the Second Derivative Test fails (i.e., f''(c) = 0)? Does this mean there is no maximum or minimum?
No, a failed Second Derivative Test (where f''(c) = 0) does not automatically mean there is no maximum or minimum. It simply means the test is inconclusive and provides no information about the point's nature. In such cases, as per the CBSE 2025-26 syllabus guidelines, you must revert to the First Derivative Test. By checking the sign of f'(x) on either side of the critical point 'c', you can determine if it's a local maximum, local minimum, or a point of inflection (like for the function y = x³ at x=0).
7. Can you explain a real-world example of how finding local maxima and minima is useful?
A classic application is in business to maximise profit. Suppose a company creates a function P(x) that models the profit for selling 'x' units of a product. By using calculus to find the derivative P'(x), setting it to zero, and solving for 'x', the company can find the critical points. The Second Derivative Test can then confirm which value of 'x' yields the maximum profit (a local maximum). This allows the business to determine the optimal production level to avoid overproduction (decreasing profits) or underproduction (missed profit opportunities).
8. What is a 'critical point' in the context of finding maxima and minima?
A critical point of a function f(x) is a point 'c' in its domain where either:
- The first derivative is zero, i.e., f'(c) = 0. These points correspond to a horizontal tangent on the graph.
- The first derivative is undefined. This can occur at sharp corners or cusps on the graph.
9. How does a point of inflection differ from a local maximum or minimum?
A point of inflection is fundamentally different from an extremum. A local maximum or minimum is a point where the function's rate of change (the first derivative) changes sign. In contrast, a point of inflection is where the function's concavity changes—from concave up to concave down, or vice versa. This means the sign of the second derivative, f''(x), changes at this point. While the slope might be zero at an inflection point (like in y = x³), it doesn't have to be. The key distinction is the change in concavity, not the switch from increasing to decreasing.
10. How is the concept of the slope of a tangent line related to identifying local maxima and minima?
The relationship is direct and foundational. The derivative of a function at a point gives the slope of the tangent line at that point. At the peak of a smooth curve (a local maximum) or the bottom of a valley (a local minimum), the curve flattens out momentarily. At these exact points, the tangent line becomes perfectly horizontal. A horizontal line has a slope of zero. This is precisely why we set the first derivative, f'(x), equal to zero to find the x-coordinates of potential local maxima and minima.
