QPaperGen

We are given:
\[
f(x) = \sin 2x
\Rightarrow f'(x) = 2\cos 2x
\]

Set the derivative to zero:
\[
f'(x) = 0 \Rightarrow \cos 2x = 0
\Rightarrow 2x = \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \frac{7\pi}{2}
\Rightarrow x = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}
\]

Now evaluate \( f(x) \) at these critical points:
\[
f\left( \frac{\pi}{4} \right) = \sin\left( \frac{\pi}{2} \right) = 1 \]
\[f\left( \frac{3\pi}{4} \right) = \sin\left( \frac{3\pi}{2} \right) = -1 \]
\[f\left( \frac{5\pi}{4} \right) = \sin\left( \frac{5\pi}{2} \right) = 1 \]
\[f\left( \frac{7\pi}{4} \right) = \sin\left( \frac{7\pi}{2} \right) = -1
\]

Also check endpoints:
\[
f(0) = \sin 0 = 0, \quad f(2\pi) = \sin(4\pi) = 0
\]

Thus, the absolute maximum value of \( f(x) \) on \([0, 2\pi]\) is
\(
\boxed{1}
\quad \text{at} \quad x = \frac{\pi}{4} \text{ and } x = \frac{5\pi}{4}
\)