Local 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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1. (0,3)

2. (3,0)

3. (1,4)

4. (4,1)

2. At which coordinates, function is decreasing in the below graph?

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1. (4, ∞ )

2. (-4, ∞)

3. (-∞, 4)

4. 4 < x < 6