Find the absolute maximum and minimum values of the following function on the given interval. Then graph the function.
\[
g(x)=2 \csc x, \frac{\pi}{6} \leq x \leq \frac{5 \pi}{6}
\]

Step 1 ：Define the function \(g(x) = 2 \csc(x)\), which is equivalent to \(g(x) = 2/\sin(x)\).

Step 2 ：Identify the interval of interest as \([\frac{\pi}{6}, \frac{5\pi}{6}]\), within which the function is defined and continuous.

Step 3 ：Find the critical points of the function within the interval. These are the points where the derivative of the function is zero or undefined.

Step 4 ：Calculate the derivative of \(g(x) = 2\csc(x)\) to get \(g'(x) = -2\csc(x)\cot(x)\).

Step 5 ：Identify that the derivative is undefined at \(x = n\pi\) and is zero where \(\cot(x) = 0\), which is at \(x = (2n+1)\pi/2\).

Step 6 ：Within the interval \([\frac{\pi}{6}, \frac{5\pi}{6}]\), the only place where the derivative is zero is at \(x = \pi/2\).

Step 7 ：Evaluate the function at \(x = \frac{\pi}{6}\), \(\pi/2\), and \(\frac{5\pi}{6}\) and compare these values to find the absolute maximum and minimum.

Step 8 ：The maximum value of the function on the interval \([\frac{\pi}{6}, \frac{5\pi}{6}]\) is 4 and the minimum value is 2. These occur at \(x = \frac{\pi}{6}\), \(\frac{5\pi}{6}\) for the maximum and \(x = \pi/2\) for the minimum.

Step 9 ：Final Answer: The absolute maximum value of the function on the interval is \(\boxed{4}\) and the absolute minimum value is \(\boxed{2}\).