1. **State the problem:** Find the value of $x$ in the interval $0 \leq x \leq 2\pi$ where the function $y = -2 \sin\left(\frac{\pi}{2} x\right)$ attains its maximum value. 2. **Recall the properties of sine function:** The sine function $\sin(\theta)$ oscillates between $-1$ and $1$. Multiplying by $-2$ flips the sine wave vertically and scales it, so $y$ ranges between $-2$ and $2$. 3. **Find when $y$ is maximum:** Since $y = -2 \sin\left(\frac{\pi}{2} x\right)$, the maximum value of $y$ is $2$, which occurs when $\sin\left(\frac{\pi}{2} x\right) = -1$. 4. **Solve for $x$:** We need to solve $$\sin\left(\frac{\pi}{2} x\right) = -1$$ The sine function equals $-1$ at angles of the form $$\frac{\pi}{2} x = \frac{3\pi}{2} + 2k\pi, \quad k \in \mathbb{Z}$$ 5. **Find $x$ values in $[0, 2\pi]$:** Substitute and solve for $x$: $$x = \frac{3\pi/2 + 2k\pi}{\pi/2} = \frac{3\pi/2}{\pi/2} + \frac{2k\pi}{\pi/2} = 3 + 4k$$ 6. **Check values for integer $k$ to be in $[0, 2\pi]$:** Since $2\pi \approx 6.283$, possible $x$ values are: - For $k=0$, $x=3$ (which is within $[0, 6.283]$) - For $k=-1$, $x=-1$ (not in interval) - For $k=1$, $x=7$ (not in interval) 7. **Conclusion:** The maximum value $y=2$ occurs at $x=3$. **Final answer:** $$\boxed{3}$$