Chapter 2.3, Problem 51AYU

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.

55.

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 51AYU

From the graph, the following results can be derived:

a. There is no absolute maximum and absolute minimum point for the given function.

b. There is no local maximum and local minimum points for the given function.

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 the point which is disconnected.

   There is no local maximum point.

   As the curve is disconnected there is no 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 the point which is disconnected.

   There is no local minimum point.

   As the curve is disconnected there is no absolute minimum.

b. From the absolute maximum and absolute minimum values, identify the local extrema that there is no local extrema points.