Q 45. - 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 $f$ 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 $[1,5]$ as its domain.

We can see from the graph that the given function has two maximum values in the interval $[1,5]$.

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

The largest value of $f$ is $f\left(1\right)=4$.

Therefore, 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 $[1,5]$.

$$\begin{gathered} f\left(2\right)=2 \\ f\left(5\right)=1 \end{gathered}$$

The smallest value of $f$ is $f\left(5\right)=1$.

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