Determine the location and value of the absolute extreme values of $f$ on the given interval, if they exist.
\[
f(x)=\sin 3 x \text { on }\left[-\frac{\pi}{4}, \frac{\pi}{3}\right] .
\]

Step 1 ：First, we need to find the critical points of the function \(f(x) = \sin 3x\). The critical points are where the derivative of the function is zero or undefined. The derivative of the function is \(f'(x) = 3\cos 3x\).

Step 2 ：Solving \(f'(x) = 0\) gives us the critical points \(x = \frac{\pi}{6}\) and \(x = \frac{\pi}{2}\). However, \(x = \frac{\pi}{2}\) is not within the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{3}\right]\), so we only consider \(x = \frac{\pi}{6}\).

Step 3 ：Next, we evaluate the function at the critical point \(x = \frac{\pi}{6}\) and the endpoints of the interval \(x = -\frac{\pi}{4}\) and \(x = \frac{\pi}{3}\).

Step 4 ：The function values at the critical point and the endpoints are approximately \(1.0\), \(-1.0\), \(-0.707\), and \(0\).

Step 5 ：The absolute maximum value is \(1.0\) and it occurs at \(x = \frac{\pi}{6}\). The absolute minimum value is \(-1.0\) and it occurs at \(x = \frac{\pi}{2}\), but this point is not within the given interval. So, within the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{3}\right]\), the absolute minimum value is \(-0.707\) and it occurs at \(x = -\frac{\pi}{4}\).

Step 6 ：Final Answer: The absolute maximum value of \(f\) on the interval \(\left[-\frac{\pi}{4}, \frac{\pi}{3}\right]\) is \(1.0\) and it occurs at \(x = \frac{\pi}{6}\). The absolute minimum value is \(-0.707\) and it occurs at \(x = -\frac{\pi}{4}\). So, the location and value of the absolute extreme values of \(f\) on the given interval are \(\boxed{\left(\frac{\pi}{6}, 1.0\right)}\) and \(\boxed{\left(-\frac{\pi}{4}, -0.707\right)}\).