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.

Short answer

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

Step-by-step solution

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$1$.