Chapter 2.3, Problem 50AYU

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

54.

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 50AYU

From the graph, the following results can be derived:

a. Absolute maximum is 4 but there is no absolute minimum.

b. Local maxima of the function $f$ is at $x=2$ and the value $f\left(2\right)=4$, and the local minimum point at $x=0$ and the corresponding values is $f\left(0\right)=2$.

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. 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 maximum points $x=\text{ 2}$ and the corresponding values are $f\left(2\right)=4$.

   The values of the local maximum at $x=\text{ 2}$ is 4.

   The value of $f$ at each endpoint of $\left(-1,3\right)$that is, $f(-1)=\text{ 3}$ and the other end point is at minus infinity.

   Therefore, the absolute maximum is same as the local maximum.

   The absolute maximum of the function $f$ is 4.

   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 $x=0$ and the corresponding values is $f\left(0\right)=2$.

   The value of $f$ at each endpoint of $\left(-1,3\right)$that is, $f\left(-1\right)=3$ and the other end point is at minus infinity.

   Therefore, there is no absolute minimum for the function $f$.

b. From the absolute maximum and absolute minimum values, identify the local extrema that the values of the local maximum at $x=2$ is $f\left(2\right)=4$ and the local minimum point at $x=0$ and corresponding value is $f\left(0\right)=2$.