Q 47. - Precalculus Enhanced with Graphing Utilities

Question

In Problems 45–52, for each graph of a function $y=f\left(x\right)$, find the absolute maximum and the absolute minimum, if they exist.

Step-by-Step Solution

Answer

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

Step 1. Given information.

The given graph of the function $y=f\left(x\right)$ is:

Step 2. Use the concept of absolute maximum and absolute minimum.

Let denote a function defined on some interval I. If there is a number u in I for which $f\left(x\right)\le f\left(u\right)$ for all x in I, then $f\left(u\right)$ is the absolute maximum of $f$ and I.

If there is a number v in I for which $f\left(x\right)\ge f\left(v\right)$ for all x in I , then $f\left(v\right)$ is the absolute minimum of $f$ on $I$.

Step 3. Find the absolute maximum.

The given graph has the closed interval $[0,4]$ as its domain.

We can see from the graph that the given function has maximum value in the interval $[0,5]$ is:

$f\left(3\right)=4$

The largest value of $f$ is $f\left(3\right)=4$.

Therefore, the absolute maximum of the function is $4$.

Step 4. Find the absolute minimum.

We can see from the graph that the given function has two minimum values in the interval $[0,4]$.

$$\begin{gathered} f\left(1\right)=1 \\ f\left(4\right)=3 \end{gathered}$$

The smallest value of $f$ is $f\left(1\right)=1$.

Therefore, the absolute minimum of the function is $1$.