How to Find Global Maxima and Minima

Do you ever wonder how to find the highest or lowest points of something? Maybe you're trying to figure out the best temperature for baking cookies or the shortest route to school. In the world of mathematics, we call these high points "global maximums" and low points "global minimums." They're like the peaks and valleys of a roller coaster, showing us the highest and lowest points along the ride.

In this article, we'll take a journey into the world of finding global maximums and minimums.

What is Global Maxima and Minima?

The global maxima also called the absolute maxima is the highest value in the entire domain of the function. The global minima also called the absolute minima is the lowest value in the entire domain of the function.

Global Maxima Definition

A function f(x) with domain D is called global maximum at x = a where a ∈ D, if f(x) ≤ f(a) for all x ∈ D. The point a is called the point of global maxima of the function and f(a) is called as the global maximum value.

Condition for Global Maxima

Condition for global maxima is given by:

x = a is point of global maxima when

f(x) ≤ f(a) for all x ∈ D

f(a) is called the global maximum value.

Global Minima Definition

A function f(x) with domain D is called global minimum at x = a where a ∈ D, if f(x) ≥ f(a) for all x ∈ D. The point a is called the point of global minima of the function and f(a) is called as the global minimum value for global minima.

Condition for Global Minima

Condition for global minima is given by:

x = a is point of global minima when

f(x) ≥ f(a) for all x ∈ D

f(a) is called the global minimum value.

Locations of Global Maxima and Global Minima

The locations of the global maxima and global minima is the maximum and minimum value in the graph respectively. The global maxima is the highest point on the graph of the function whereas the global minima is the lowest point on the graph of the function. There can only be one global maxima and one global minima of any function.

Understanding Critical Points in Multivariable Functions

To find the critical points in multivariate functions we follow the below steps.

• First, find the first order partial derivative of the function w.r.t all the given variable separately.

• Then, equate each partial order derivative to get the values of the variable.

• Combine all the obtained values to get the critical points of multivariate functions.

Diagrammatic Representation of Global Maxima and Minima

The below is the diagrammatic representation of global maxima and minima.

👁 Absolute-Maxima-and-Minima

How to Find Global Maxima and Minima

We can find the global maxima and global minima in different ways:

• Global Maxima and Minima in Closed Interval

• Global Maxima and Minima in Entire Domain

• First Derivative Test for Maxima and Minima

• Second-Order Derivative Test for Maxima and Minima

Global Maxima and Minima in Closed Interval

Below are the steps to find the global maxima and global minima in closed interval.

• First, find the first derivative of the given function i.e., f'(x).

• Then, find the critical points of the function in the interval D by putting f'(x) = 0

• Then, find the value of the function at the extreme points of D interval.

• The highest and the lowest value obtained in the above steps is called as the global maximum and global minimum of the function.

Global Maxima and Minima in Entire Domain

Below are the steps to find the global maxima and global minima in entire domain.

• First, find the first derivative of the given function i.e., f'(x).

• Then, find the critical points of the function by putting f'(x) = 0

• Then, find the value of the function at the extreme points.

• Then, check whether the value of function when x →∞ or x→-∞. Also check the discontinuity points.

• The global maximum and the global minimum for function in its entire domain is given by maximum and minimum of all these values.

First Derivative Test for Maxima and Minima

Examining the First Derivative Test for Maxima and Minima When analyzing a function's first derivative, we observe the slope of the function. Near a maximum point, the slope ascends towards the maximum point, reaches zero at that point, and then descends as we move away. Similarly, near a minimum point, the slope decreases towards the minimum point, attains zero, and then ascends away from it. This information aids in determining whether a point is a maximum or minimum.

Consider a function ? which is continuous at a critical point, defined in an open interval ?, and ?′(?) = 0 (indicating zero slope at ?). Then, by inspecting the nature of ?′(?) to the left and right of ?, we can classify the point as follows:

• Local Maximum: If ?′(?) changes sign from positive to negative as ? increases through ?, then ?(?) gives the maximum value of the function in that range.

• Local Minimum: If ?′(?) changes sign from negative to positive as ? increases through ?, then ?(?) gives the minimum value of the function in that range.

• Point of Inflection: If the sign of ?′(?) doesn't change as ?x increases through ?, and ? is neither a maximum nor minimum of the function, then ? is a point of inflection.

Second-Order Derivative Test for Maxima and Minima

In the second-order derivative test, we first examine the function's first derivative. If it evaluates to zero at the critical point ?=? (?′(?) = 0), we then analyze the second derivative of the function. If the second derivative exists within the specified range, we determine the point as follows:

• Local maximum: If ?′′(?) < 0

• Local minimum: If ?′′(?) > 0

• Test Inconclusive: If ?′′(?) = 0

Read More:

• Absolute Maxima and Minima
• Maxima and Minima

Solved Examples on How to Find Global Maxima and Minima

Example 1: Find the global maxima and minima value of function f(x) = 2x² - 4x in the interval [0, 2].

Solution:

f(x) = 2x² - 4x

First find f'(x)

f'(x) = 4x - 4

Now put f'(x) = 0

4x - 4= 0

4x = 4

x = 1

There is only one critical point of function i.e., x = 1

Now find the value of f(x) at critical point

At x = 1 f(1) = - 2

Now find the value of f(x) at extremum points I.e.,

x =0 f(0) = 0

x = 2 f(2) = 0

So, f(x) has its global maxima value 0 at x= 0 and x = 2 and global minima value -2 at x = 1.

Example 2: Find the global maxima and minima of function f(x) = 4e^x + 3 in the interval [0, 3].

Solution:

f(x) = 4e^x + 3

First find f'(x)

f'(x) = 4e^x

The above equation will never be zero for any value of x within the interval. So, we find values of f(x) at extremum points. Now find the value of f(x) at extremum points I.e.,

x =0 f(0) = 7

x = 3 f(3) = 4e³ + 3

So, f(x) has its global maxima value 4e³ + 3 at x= 0 and global minima value 7 at x = 0.

Example 3: Find the global maximum value and minimum value of function f(x) = 1 / (x - 2) in its entire domain.

Solution:

The domain of the function f(x) = 1 / (x - 2) is R - {2}.

f'(x) = -1 / (x - 2)²

There is no point in function domain where f'(x) = 0

So, we check for the values of function when x tends to ∞

• x→-∞ ⇒ f(x) → 0
• x→∞ ⇒ f(x) → 0
• x→ 2 ⇒ f(x) → ∞

Therefore, there is no maximum value and minimum value exists when x→-∞ or ∞ ⇒ f(x) → 0.

Practice Problems on How to Find Global Maxima and Minima

P1: Find the global maxima and minima of function f(x) = x⁴ - x in the interval [1, 4].

P2: Find the global maxima and minima value of function f(x) = 1 / (x -5) in its entire domain.

P3: Find the global maximum and minimum value of function f(x) = 4x³ + 7x² + 2x + 1 in the interval [0, 3].

P4: Find the global maxima and minima of function f(x) = 4cos x + sin x in the interval [0, 1].