Q 46. - 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 $0$.

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 $[0,5]$ as its domain.

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

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

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

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 $[0,5]$.

$$\begin{gathered} f\left(1\right)=1 \\ f\left(5\right)=0 \end{gathered}$$

The smallest value of $f$ is$f\left(5\right)=0$.

Therefore, absolute maximum of the function is $0$.