Q 50. - 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 function has no absolute minimum.

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 is the absolute maximum of $f$ and I.

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

Step 3. Find the absolute maximum.

We can see from the graph that the given function has the domain $\{x\overline{)-1\le x<3\}}$.

We can see from the graph that the given function has maximum value $f$ on its domain is:

$f\left(2\right)=4$

Therefore, the absolute maximum of the function is 4.

Step 4. Find the absolute minimum.

The function is approaching infinity at point $x=3$.

Thus, the largest value of $f$ is not defined.

The function has no absolute minimum value.