Chapter 2.3, Problem 45AYU

In Problem 49-56, for each graph of a function $y\text{ = }f\left(x\right)$, find the absolute maximum and the absolute minimum, if they exist. Identify any local maximum values or local minimum values.

49.

To determine

To find: The following values using the given graph:

a. Absolute maximum and minimum if they exist.

b. Local maximum and minimum values.

Answer to Problem 45AYU

From the graph, the following results can be derived:

a. The absolute maximum is 4 and the absolute minimum is 1.

b. Local maxima of the function $f$ is at $x=\text{ 3}$ and the value $f\left(3\right)=3$, also the local minima of the function $f$ is at $x=\text{ 2}$ and the value $f\left(2\right)=2$.

Explanation of Solution

Given:

It is asked to find the absolute maximum and minimum of the given function and also identify its local maximum and minimum values.

Graph:

Interpretation:

a. Absolute maximum: The absolute maximum can be found by selecting the largest value of $f$ from the following list:

1. The values of $f$ at any local maxima of $f$ in $\left[a,b\right]$.
2. $\left[a,b\right]$-that is, $f\left(a\right)$ and $f\left(b\right)$.

   It can be directly concluded from the graph and the definition that the curve has local maximum point at $x=\text{ 3}$.

   The values of the local maximum at $x=\text{ 3}$ is 3. Therefore, the local maximum point is $\left(3,\text{ 3}\right)$.

   The value of $f$ at each endpoint of $\left[1,\text{ 4}\right]$ and $\left[5,\text{ 1}\right]$-that is, $f\left(1\right)=\text{ 4}$ and $f\left(5\right)=\text{ 1}$.

   The largest of these, 4, is the absolute maximum.

   Absolute minimum: The absolute minimum can be found by selecting the smallest value of $f$ from the following list:

3. The values of $f$ at any local minima of $f$ in $\left[a,b\right]$.
4. The value of $f$ at each endpoint of $\left[a,b\right]$-that is, $f\left(a\right)$ and $f\left(b\right)$.

   It can be directly concluded from the graph and the definition that the curve has local minimum point at $x=\text{ 2}$.

   The values of the local minimum at $x=\text{ 2}$ is 2. Therefore, the local minimum point is $\left(2,\text{ 2}\right)$.

   The value of $f$ at each endpoint of $\left[1,\text{ 4}\right]$ and $\left[5,\text{ 1}\right]$-that is, $f\left(1\right)=\text{ 4}$ and $f\left(5\right)=\text{ 1}$.

   The largest of these, 1, is the absolute minimum.

b. From the absolute maximum and absolute minimum values, identify the local extrema that is the local maxima point is at $x=\text{ 3}$, the value is $f\left(3\right)=\text{ 3}$ and the local minima point is at $x=\text{ 2}$, the value is $f\left(2\right)=\text{ 2}$.