Answer
The absolute maximum $g(\pi/3)=g(2\pi/3)=2\sqrt 3/3$ and the absolute minimum $g(\pi/2)=1$.
See graph.
Work Step by Step
Step 1. Given the function $g(x)=csc(x), \pi/3\leq x\leq 2\pi/3$, we have $g(\pi/3)=2\sqrt 3/3, g(2\pi/3)=2\sqrt 3/3$,
Step 2. $g'(x)=-csc(x)cot(x)$, let $g'(x)=0$, we get critical points at $x= \pi/2$ and $g(\pi/2)=1$
Step 3. We can identify the absolute maximum as $g(\pi/3)=g(2\pi/3)=2\sqrt 3/3$ and the absolute minimum as $g(\pi/2)=1$ on the given interval.
Step 4. See graph.