Answer
The function has an absolute minimum at $x=0$ but does not have any absolute maximum.
Work Step by Step
$$f(x)=|x|\hspace{2cm}-1\lt x\lt2$$
The graph is sketched below.
- The graph obviously has an absolute minimum value, which is $f(0)=0$.
- The graph, however, does not have an absolute maximum value.
The graph does seem to reach the highest point at $x=2$, but since we consider here the interval $(-1,2)$, we do not include the point where $x=2$. And since there are no other maximum values, the graph does not have any absolute maximum in the defined interval.
This answer is still consistent with Theorem 1, because Theorem 1 requires that the function $f$ in question be examined in a CLOSED interval, not an open one like in this case.
