Q 48. - 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 minimum is 1 and there is no absolute maximum. 

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.

We can see from the graph that the given function has the domain $\left\{x\overline{)0\le x<3,x\ne 2}\right\}$.

Here we are excluding 2 from the domain because of the "hole" at $(2,4)$.

There is no absolute maximum.

The reason being, as we trace the graph getting closer to the point $(2,4)$, there is no single largest 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:

$f\left(0\right)=1$

The smallest value of $f$ is $f\left(0\right)=1$.

Therefore, the absolute minimum of the function is 1.