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# Minimum|Definition & Meaning

**Definition**

The **minimum** is the **smallest value** that can be possible in a given **set, function,** or any other **mathematical** set of values. Furthermore, a minimum is possible between a **bounded** function whose upper and lower limits are clearly defined. It is usually depicted as a **min f(x)** or **min {a}, **where **f(x) **is a **mathematical function** and** {a}** is a set of values given, and the word min represents finding the lowest value of such functions and sets.

Figure 1: A Figure showing the minimum value of a function y = x$^2$ + x

The **minimum** is a part of the **broader** definition of **extrema.** The word **“extrema”** explains the **upper** and **lower** values of the given functions, which are possible within a set of **constraints** on that given function. **Maximum** or **maxima** here is similar to the **minima,** but it contains all the **highest** or **largest values** possible for the **function** to **map** or the set to contain.

In a mathematical function, there are **two types** of **minima: Local Minima** and **Absolute Minima.** These two quantities explain the function depicted as a curve on a graph and help find the **implicit meaning** and **trends** of the **functions.** For a set, there is always a **single minimum** or maximum value, as it has a set of **predefined** values.

These predefined values cannot be **altered** or **modified** by any **external factor.**

## Local Minimum and Global Minimum

Two types of **minimums** shape the function and explain its implicit meanings: **Local minimum** and **global minimum.** These two types of minimum help in **clearly defining** the properties of a function or a set and explain the trend that the set of values is **depicting.**

A **local minimum** is the minimum value of a function for a **specific interval** of a **function.** The local minima are **not necessarily** the **lowest value** of all the function’s intervals, but they will be the lowest value for the **interval** where it occurs. Furthermore, we can also explain this as the lowest point concerning the **nearby points** on the **graph** is known as the Local Minimum.

The second type of minimum, the **Global Minimum,** is the **lowest value** of a function for the **whole function.** This means that **no other point** on the function is **lower** than the **global minimum.** This way, we can identify what is the lowest point that the function can reach **throughout** its **whole interval.**

Figure 2: A curve showing different local minima and a global minimum.

## Finding the Minimum Value of a Function

For a given function, it is a simple process to **find** the maxima or the minima value of a function. This extreme value is also known as a **stationary value** on the function. This means that the function is having a **zero change** in the function **f(x)** as the value of the **x** changes.

Thus, all we need to do is to find the **derivative** of the **function** given and **equate it** to **zero.** This way, the value of **x** in the function **f(x)** will be the stationary point, where there is no change in the value of f(x) for a **minute change** in** x**.

The following is the formula to find the stationary point for which we can find the minimum value of the function.

\[\frac{\mathrm{d}}{\mathrm{d}x}f(x) = 0\]

So after **finding** the **stationary point,** all you need is to find the **double derivative** of the **function** **f(x)** and **enter** that **stationary point x** into the **equation.**

If the double derivative’s result is **greater** than zero, it will be a **minimum point,** and if the result is **less** than zero, it will be a **maximum.** This way, we can find the minimum of a function.

## Applications and Significance of Maxima and Minima

Maxima and minima have innumerable **applications** in **real-life problems** and help in solving many different **dilemmas** that **professionals** face in their daily lives. An **economist** can utilize the maximum and minimum values of the total **profit** function to gain a sense of the **salary ceilings** that the business must adhere to avoid going bankrupt.

An **engineer** can utilize a function’s **maximum** and **lowest** values to identify the **boundaries** of the function in practical situations. If you can find a use for a **train’s speed,** for instance, knowing the train’s **top speed** can help you choose **materials** that will be **durable** enough to withstand the **strain** brought on by such **high speeds** and can be used to make **brakes, rails,** and other components that will allow the train to run **smoothly.**

A **doctor** can use the **maximum** and **lowest** values of the function defining the total **bilirubin level** in the **bloodstream** to estimate the dose that has to be given to various patients to return their bilirubin levels to normal. Hence, this can cure patients that face **Jaundice disease** early on and much more effectively.

A **power system engineer** can use the **maximum** and **minimum constraints** of a **generator** and **loads** to determine the **scheduling** of different generators that will give the most **optimum** value of the **cost** of **electricity** generated to ensure the **lowest** possible **cost** of **generation.**

Thus, the maxima and minima play a large role in helping the world become much more **efficient** and **progressive** toward a more **sustainable** outcome.

## An Example Illustrating the Usage of Minimum in a Function

We are given a **polynomial function** written as below:

\[ f(x) = x^4-3x^3 + x^2 + x \]

**Draw** the above function on a **graph** and denote the **Local** and **Global minima** of the **respective function.**

**Solution**

As this is a **polynomial** of **order four,** it will have **three extreme points.** Now we have to identify, on the graph, the minimum points, and their values. Below is the figure that is plotted using GeoGebra for the given function:

Figure 3: A curve showing Local and Global Minima for the function $ f(x) = x^4-3x^3 + x^2 + x $

Hence, according to the curve, we can see we have two **minima,** occurring at **A, x = -0.2322,** and** C,** at** x = 1.922. **

As the value at **A, x= 1.922**, is **-2.0377** and at **C**, **x** **=** **-0.2322,** is **-0.137, point C** is the **global minimum,** whereas **point A** is the **local** **minimum.**

*All the mathematical figures and graphs were created using GeoGebra.*