Local Maxima and Minima

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. 

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:

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. 

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. 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 either towards 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 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 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

Nature of f’(a)

Nature of Slope

Example

Local Extremum

f’(a) > 0

Positive

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Neither

f’(a) < 0

Negative

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Neither



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Local Maximum

f’(a) = 0

Zero

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Local Maximum



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Neither



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Local Minimum

Not defined

Not Defined

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Local Maximum



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Neither


Solved Examples

1. Determine the Local Maxima and Minima for the Function y = x³ - 3x + 2

1. 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.

Check the Below Graph to Verify the Calculations.

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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 = \[\sqrt{3}\]/3 .By initially looking at x = \[\sqrt{3}\]/3, we can see that f(\[\sqrt{3}\]/3) = -2\[\sqrt{3}\]/9. Now, we will check two points placed at either side of x = \[\sqrt{3}\]/3 by ensuring that no value is far away from the critical value. As,\[\sqrt{3}\] < 3 and \[\sqrt{3}\]/3 < 1 and we can make use of x = 0 and x=1. As f(0) = 0 > -2 \[\sqrt{3/9}\] and f(1) = 0 > -2\[\sqrt{3/9}\], there should be a local minimum at x =(\[\sqrt{3}\]/3 ) . For x =-(\[\sqrt{3}\]/3 ), we can see that f - (\[\sqrt{3}\]/3 ) = 2\[\sqrt{3}\]/9. For this, we will use x = 0  and x= -1and we will determine f(-1) = f(0) = 0 < 2\[\sqrt{3}\]/9, so there should be local maxima value at x = - (\[\sqrt{3}\]/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

FAQ (Frequently Asked Questions)

1. What are the Different Applications of Maxima and Minima?

Some of the applications of maxima and minima are given below:

For an Engineer- The maximum and the minimum values of a function can be used to place its limits in real-life. For example, if you can determine an adequate function for the speed of a car then determining the maximum possible speed of the train can enable you to select the equipment that would be sufficient enough to resist the pressure because of such high speeds and accordingly it will be helpful for them to design the brakes and the wheels etc. for the car to run smoothly.

For an Economist - The maximum and the minimum values of the total profit function helps to get an idea of boundaries, the company must set on the salaries of the employees, so that the company does not face losses.

For a Doctor -The maximum and the minimum values of the function outline the total blood level in the bloodstream and accordingly helps the doctor to determine the dosage should be suggested to different patients to bring their blood pressure levels back to normal.

2. Explain Absolute Maxima and Absolute of a Function?

If (a,b) is a point in the region D in the xy-plane, then Z = f(x,y) is said to have

  • An absolute Maxima in the region D at (p,q) when f(x,y) ≤ f(p,q) holds for all (x,y) in D.

  • An absolute Maxima in the region D at (p,q) when f(x,y) ≥ f(p,q) holds for all (x,y) in D.


Absolute maxima and absolute minima are also known as global maxima and minima.

For the function of a single variable, when f(x,y) is continuous on D, an absolute maxima and absolute minima always appears somewhere on D

Steps to Find the Absolute Maxima and Absolute Minima:

  1. Find the critical points on the interior. It  involves the computation of f(x,y)and setting it equals to 0.

  2. Maxima or minimize f(x,y) on the boundary.

  3. Compare the values of f(x,y) that you received from the above first 2 steps. The biggest value will be the absolute maxima and the smallest value will be the absolute minima.