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**Maximum|Definition & Meaning**

**Definition**

**Mathematics** defines a **maximum** as the **highest value** of a **set**. In other words, it is the **highest value** that a **certain quantity** can **attain**. The concept of maximum is often used in **optimization** **problems** where the **goal** is to **find** the **best** **solution** given certain constraints. It is often used to find the maximum value of a function or equation.

**Conceptual Overview **

**Maximum of Function**

Figure 1 – Maximum Age among four persons

A **function** can be **maximized** by finding the** highest** possible **value** with the concept of maximum. Using an **input (called the independent variable)** and a set of instructions, a function **produces** **an output (called the dependent variable)**. A **function** is often **represented** by a **graph,** and its inputs and outputs are linked by a mathematical rule.

Consider an **example** in which the** age** of the **4 persons** are given **Elon** is **5 years old**, **Bill** is **15 years old**, **Harry** is **12 years old** and **Parker** is **13 years old** we will **convert** the ages into set form **{5,12,13,15}**, when we try to find the **maximum** of the set i.e. **max{5,12,13,15}=15**. So the maximum age among the **persons** will be** 15**.

**Steps to Find Maximum of a Function**

Figure 2 – Maximum of a function

We must **first determine** the **domain** of the **function** in order to **find** its **maximum value**. An **independent variable’s** domain **consists** of all **possible values** it can assume. **Time**, for instance, might be the **independent variable**, and all real numbers over or equal to zero might be its domain.

We can **determine** the **maximum value** in a **function** by **taking** its **derivative** and **setting** it to **zero**. **Once** we **determine** the** domain**, we can **calculate** the **maximum value**. An **input changes** and the **output changes** accordingly, so the **derivative measures** how the** output changes**. By** setting** the derivative equal to **zero**, we are **finding** the **points** at which the **output** of the function is **not changing**, which means that it is **either** at a **maximum** or a **minimum** value.

**Alternate Method to Find the Maximum of a Function**

Figure 3 – Alternate method to find maximum of function

However, we can also look at the **function’s graph** to determine the maximum value. The **highest value** is **found** **at** the** apex** of the **upward-opening parabola** that forms the graph in the case of $f(x) = x^{2}$. The **coordinates** of the **parabola’s vertex** must be **discovered** in order **to determine** this **maximum value**. By **averaging** the **x-coordinates** of the **two sites** where the derivative equals zero, the vertex—the point at which the parabola changes direction—can be identified. Since there are **no points** in this scenario **where** the **derivative equals zero**, the **vertex** is **at (0, 0)**.

**Properties of Maximum**

There are several properties that are often associated with the concept of maximum in mathematics:

**Uniqueness**

Most of the time, a **particular set** or range only **contains one maximum value**. For instance, there will typically be just one value that is the **highest** if we are looking for the greatest value of a function inside a particular range.

**Existence**

There must be at least **one maximum value** for a given set or range. This is because there must be **at least one value** in the set or range for there to be a maximum, and the **maximum value** is just the highest value within the set or range.

**Monotonicity**

Figure 4 – Monotonicity of function

As the **input** values **rise**, the **maximum value** of a **function** either **rises** or stays the **same**. This means that if we are looking for the maximum value of a function, we may frequently focus our search by only taking into account **input values greater than** the **maximum value** at the moment.

**Continuity**

Figure 5 – Continuity of function

A **function’s maximum value** is oft**en a continuous point** included inside the set or range. This indicates that the function’s **value won’t** drastically **change** at its maximum point.

**Global vs. Local Maximum**

Figure 6 – Global vs Local Maxima

The **greatest value** inside the **entire set** or range is referred to as a **global maximum**, while the **highest value** within a smaller subset of the set or range is known as a **local maximum**.

**Boundaries**

At the **edges** of the set or range, a **function’s maximum value** may **occur**. For instance, if we are attempting to determine a function’s maximum value between **[0, 1]**, the **greatest value** may occur **at** either** x = 0 or x = 1**.

**Illustration of Maximum with Example**

Figure 7 – Example of Maximum of a function

Here is an **example** of how the concept of **maximum** might be used in mathematics:

**Suppose** we have a** function** $f(x) = x^{2}-4x + 3$. We want to **find** the **maximum value** of this function. To do this, we can **take** the **derivative** of the **function** and **set** it equal to **zero**:

$f'(x) = 2x-4 = 0$

Solving this equation for **x gives us x = 2**. This means that the **maximum** **value** of the function **occurs** when** x = 2**.

To find the **maximum** value of the function, we can then **evaluate** the **function** at **x = 2**:

$f(2) = 2^{2}-4(2) + 3 = 4-8 + 3 = -1$

Thus, the **maximum** value of the **function** is **-1.**

Alternatively, we could also find the **maximum** **value** of the function by **looking** at the **graph** of the **function**. The **graph** of **f(x)** is a **parabola** that **opens** **upwards**, and the **maximum** **value** **occurs** at the **top** of the **parabola**. To find this maximum value, we need to **find** the **coordinates** of the **vertex** of the **parabola**.

The **vertex** is the **point** at which the **parabola** **changes** **direction**, and it can be **found** **by** taking the **average** of the **x-coordinates** of the two points **at** **which** the **derivative** is **equal** to **zero**. In this case, there is only one point at which the derivative is equal to **zero (x = 2)**, so the **vertex is at (2,-1)**.

This **example** **illustrates** how the concept of **maximum** can be used in **mathematics** to find the **highest** **possible** value of a **function** or equation.

By **taking** the **derivative** of the **function** and **setting** it equal to **zero**, or by **looking** at the **graph** of the **function** and **finding** the **coordinates** of the **vertex**, we can determine the maximum value of the function and understand the limits and boundaries of the quantity that it represents.

*All mathematical drawings and images were created with GeoGebra.*