Introduction

When we deal with functions, we are often interested in their largest or smallest values.

  • The global maximum of a function is the largest y -value that the function attains on its domain. There can only be (at most) one global maximum. It's also called the absolute maximum.

  • The global minimum of a function is the smallest y -value that the function attains on its domain. There can only be (at most) one global minimum. It's also called the absolute minimum.

If one or both exist, the global maximum and minimum values of a function are collectively called the global extrema of the function.

For example, in the graph below, the global maximum is 2 , while the global minimum is -2 .

The point where a function reaches its global maximum/minimum is called the global maximum/minimum point. So in this case,

  • (-2,2) is the global maximum point, and

  • (1,-2) is the global minimum point.

Watch out! The global maximum/minimum is unique. It's a value of the function ( y -coordinate on the graph). However, there can be several global maximum/minimum points. All of these will have the same y -coordinate (global maximum/minimum) but different x -coordinates.

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Example: Identifying the Global Maximum of a Function

What is the global maximum point of the function shown below?

EXPLANATION

First, we recall the following definitions:

  • The global maximum of a function is the largest value that the function attains on its domain. There can only be (at most) one global maximum.

  • A global maximum point is a point (x,y) where a function reaches a global maximum. There can be zero, one, or many global maximum points.

For our function, the maximum value reached by the function is f(x)=4. Therefore, the global maximum is 4. The function reaches this value when x=-1, so the global maximum point is (-1,4).

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Practice: Identifying the Global Maximum of a Function

What is the global maximum of the function shown above?

a
 $0$
b
 $3$
c
 $2$
d
 $1$
e
 $-2$
Practice: Identifying the Global Maximum of a Function

What is the global maximum point of the function shown above?

a
 $(2,1)$
b
 $(3,-3)$
c
 $(1,-1)$
d
 $(0,-3)$
e
 $(-0.5,-2)$
Example: Identifying the Global Minimum of a Function

What are the global minimum points of the function shown below?

EXPLANATION

First, we recall the following definitions:

  • The global minimum of a function is the smallest value that the function attains on its domain. There can only be (at most) one global minimum.

  • A global minimum point of a function is a point (x,y) where a function reaches a global minimum. There can be zero, one, or many global minimum points.

For our function, the minimum value reached by the function is f(x)=0. Therefore, the global minimum is 0. The function reaches this value when x=-3 and when x=1, so the global minimum points are (-3,0) and (1,0).

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Practice: Identifying the Global Minimum of a Function

What is the global minimum of the function shown above?

a
 $1.5$
b
 $-2$
c
 $-1.5$
d
 $2$
e
 $3$
Practice: Identifying the Global Minimum of a Function

What is the global minimum point of the function shown above?

a
 $(3,0)$
b
 $(2,-4)$
c
 $(-0.5,-1)$
d
 $(3,2)$
e
 $(0,0)$
Example: Identifying Whether the Global Extrema of a Function Exist

The graph y = f(x) is shown below. Which of the following statements is true?

  1. The global maximum of f(x) is 1.
  2. The global minimum of f(x) is -4.
  3. The function does not have any global extrema.
EXPLANATION

Let's examine each statement.

  • Statement I is false. As x increases on the right-hand side of the graph, the function continues to increase without bound (as indicated by the arrow). So, the function does not reach a global maximum value.

  • Statement II is false. At first glance, it might seem that the point (-4,-4) corresponds to the global minimum, but since x=-4 is not included in the domain (as indicated by the open circle), the function does not have a global minimum.

  • Statement III is true. The function does not have a global maximum and does not have a global minimum. Therefore, the function does not have any global extrema.

Therefore, the correct answer is "III only."

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Practice: Identifying Whether the Global Extrema of a Function Exist

The graph $y = f(x)$ is shown above. Which of the following statements are true?

  1. The global maximum of $f(x)$ is $4.$
  2. The global minimum of $f(x)$ is $1.$
  3. The function does not have any global extrema.
a
 II only
b
 III only
c
 None of the statements are true.
d
 I only
e
 I and II only
Practice: Identifying Whether the Global Extrema of a Function Exist

The graph $y = f(x)$ is shown above. Which of the following statements is true?

  1. The global maximum of $f(x)$ is $2.$
  2. The global minimum of $f(x)$ is $-2.$
  3. The function does not have any global extrema.
a
 I and II only
b
 III only
c
 I only
d
 II only
e
 None of the statements are true.
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