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 given function has no absolute maximum and minimum values.

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 role="math" localid="1645895785421" $f\left(v\right)$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 $\{x\overline{)-2\le x\le 4,x\ne -1andx\ne 3\}}$.

Here we are excluding $-1$and 3 from the domain because of the holes at $(-1,-3)$and $(3,2)$.

There is no absolute maximum.

The reason being, as we trace the graph getting closer to the point $(3,2)$, there is no single largest value.

The function has no absolute maximum value.

Step 4. Find the absolute minimum.

There is no absolute minimum.

The reason is, as we trace the graph getting closer to the point $(3,2)$, there is no single smallest value.

The function has no absolute minimum value.