Maxima and Minima

As the name suggests, this topic is devoted to the method of finding the maximum and the minimum values of a function in a given domain. It finds application in almost every field of work, and in every subject. Let’s find out more about the maxima and minima in this topic.

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Some day-to-day applications are described below:

• To an engineer – The maximum and the minimum values of a function can be used to determine its boundaries in real-life. For example, if you can find a suitable function for the speed of a train; then determining the maximum possible speed of the train can help you choose the materials that would be strong enough to withstand the pressure due to such high speeds, and can be used to manufacture the brakes and the rails etc. for the train to run smoothly.
• To an economist – The maximum and the minimum values of the total profit function can be used to get an idea of the limits the company must put on the salaries of the employees, so as to not go in loss.
• To a doctor – The maximum and the minimum values of the function describing the total thyroid level in the bloodstream can be used to determine the dosage the doctor needs to prescribe to different patients to bring their thyroid levels to normal.

Types of Maxima and Minima

The maxima or minima can also be called an extremum i.e. an extreme value of the function. Let us have a function y = f(x) defined on a known domain of x. Based on the interval of x, on which the function attains an extremum, the extremum can be termed as a ‘local’ or a ‘global’ extremum. Let’s understand it better in the case of maxima.

Local Maxima

A point is known as a Local Maxima of a function when there may be some other point in the domain of the function for which the value of the function is more than the value of the local maxima, but such a point doesn’t exist in the vicinity or neighborhood of the local maxima. You can also understand it as a maximum value with respect to the points nearby it.

Global Maxima

A point is known as a Global Maxima of a function when there is no other point in the domain of the function for which the value of the function is more than the value of the global maxima. Types of Global Maxima:

• Global maxima may satisfy all the conditions of local maxima. You can also understand it as the Local Maxima with the maximum value in this case.
• Alternately, the global maxima for an increasing function could be the endpoint in its domain; as it would obviously have the maximum value. In this case, it isn’t a local maximum for the function.

Similarly, the local and the global minima can be defined. Look at the graph below to identify the different types of maxima and minima.

Stationary Points

A stationary point on a curve is defined as one at which the derivative vanishes i.e. a point (x0, f(x0)) is a stationary point of f(x) if $${[\frac{df}{dx}]_{x = x_0} = 0}$$. Types of stationary points:

• Local Maxima
• Local Minimas
• Inflection Points

We won’t discuss inflection points here. As of now though, you must note that all the points of extremum are stationary points.

Proof: I’ll prove the above statement for the case of a Local Maxima. Others will simply follow from this. Let us have a function y = f(x) that attains a Local Maximum at point x = x0. Near the extremum point, the curve will look something like this:

Fig 1.

Clearly, the derivative of the function has to go to 0 at the point of Local Maximum; otherwise, it would never attain a maximum value with respect to its neighbors.

The Second Derivative Test

This test is used to determine whether a stationary point is a Local Maxima or a Local Minima. Whether it is a global maxima/global minima can be determined by comparing its value with other local maxima/minima. Let us have a function y = f(x) with x = x0 as a stationary point. Then the test says:

• If $${[\frac{d^2f}{d^2x}]_{x = x_0} < 0}$$, then x = x0 is a point of Local Maxima.
• If $${[\frac{d^2f}{d^2x}]_{x = x_0} > 0}$$, then x = x0 is a point of Local Minima.
• If $${[\frac{d^2f}{d^2x}]_{x = x_0} = 0}$$, then check in the following way:
• If for x > x0, $${[\frac{df}{dx}]_{x = x_0} < 0 }$$ and for x < x0, $${[\frac{df}{dx}]_{x = x_0} > 0}$$ i.e. the function is increasing for x < x0 and decreasing for x > x0; we can conclude that x = x0 is a point of Local Maxima.
• Similarly, if for x > x0, $${[\frac{df}{dx}]_{x = x_0} > 0 }$$ and for x < x0, $${[\frac{df}{dx}]_{x = x_0} < 0}$$ i.e. the function is decreasing for x < x0 and increasing for x > x0; we can conclude that x = x0 is a point of Local Minima.

The proof of the third case can be understood by looking at Fig 1. above for local maxima. Similarly, for local minima, one can get:

Fig 2.

Proof of the Second Derivative Test

We’ll prove the test for the case of a Local Minima. The proof for a Local Maxima will follow in a similar fashion. Take a look at the Fig 2. above. One can see that the slope of the tangent drawn at any point on the curve i.e. $${\frac{dt}{dx}}$$ changes from a negative value to 0 to a positive value, near the point of local minima. T

his means that the function that is represented by(say)  $${f(x) = \frac{dt}{dx}}$$ behaves like an increasing function. The condition for a function to be increasing is: $${\frac{df}{dx} > 0}{\text{ i.e.} \frac{d^2y}{d^2x} > 0}$$ This confirms that the function will have a local minima if the first derivative is 0, and the second derivative is positive at that point.

Solved Examples for You on Maxima and Minima

Question 1 : Find the local maxima and minima for the function y = x3 – 3x + 2.

Answer : We’ll need to find the stationary points for this function, for which we need to calculate $${\frac{df}{dx}}$$. We’ll proceed as follows:

$${ y = x^3 – 3x +2}$$
$${\frac{dy}{dx} = 3x^2 – 3}$$

At stationary points, $${\frac{dy}{dx} = 0}$$. Thus, we have;

$${3x^2 – 3 = 0}$$
$${3(x^2 – 1) = 0}$$
$${(x – 1)(x + 1) = 0}$$
$${ \text{ x = 1 / x = -1}}$$

Now we have to determine whether any of these stationary points are extremum points. We’ll use the second derivative test for this:

$${\frac{dy}{dx} = 3x^2 – 3}$$
$${\frac{d^2y}{d^2x} = 6x }$$

• For x = 1; $${\frac{d^2y}{d^2x} = 6/times{1} = 6}$$, which is positive. Thus the point (1, y(x = 1)) is a point of Local Minima.
• For x = -1; $${\frac{d^2y}{d^2x} = 6/times{-1} = -6}$$, which is positive. Thus the point (-1, y(x = -1)) is a point of Local Maxima.

We can see from the graph below to verify our calculations:

This concludes our discussion on this topic of maxima and minima.

Question 2: What are relative maxima and relative minima?

Answer: Finding out the relative maxima and minima for a function can be done by observing the graph of that function. A relative maxima is the greater point than the points directly beside it at both sides. Whereas, a relative minimum is any point which is lesser than the points directly beside it at both sides.

Question 3: How to find out the absolute maxima of a function?

Answer: Finding the absolute maxima:

Firstly, find out all the critical numbers of the function within the interval [a, b].

Then, plug in every single critical number from the first step into the function i.e. f(x).

Plugin the ending points that are (a) and (b) into the function f(x).

Finally, the biggest value is the absolute maxima and the lowest value is the absolute minima.

Question 4: What is the absolute maxima?

Answer: The biggest value that a mathematical function can consume over its whole curve. The absolute maxima on the graph takes place at x = d, and the absolute minima of that graph takes place at x = a.

Question 5: What are the local and global maxima and minima?

Answer: The global maxima and minima of any function are known as the global extrema of that function. Whereas, the local maxima and minima are said to be the local extrema.

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