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 minimum is 0 and the function has no absolute maximum.

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 $f\left(x\right)\le f\left(v\right)$ 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 $\{x\overline{)-1\le x<4\}}$.

The function is approaching infinity at point $x=4$.

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

The function has no absolute maximum value.

Step 4. Find the absolute minimum.

We can see from the graph that the given function has a minimum value at its domain is:

$f\left(0\right)=0$

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

Therefore, the absolute minimum of the function is 0.