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The correct choice, which is a check valve, is essential for preventing backflow in the system. In the context of automatic central control mixing valves, check valves ensure that water flows in one direction only. This is crucial to maintain the desired water temperature and pressure, as mixing valves rely on a consistent flow of hot and cold water to achieve their function. Without check valves, there is a risk of cross-connection where hot and cold water could flow backward into the opposing supply line, potentially causing hazardous conditions or mixing of the supplies inappropriately.

The use of a relief valve, while important in other applications to prevent pressure buildup, does not serve this specific purpose of directing flow within mixing valves. Similarly, ball valves are often used for on/off control rather than for preventing backflow, and butterfly valves are usually applied in larger systems for regulating flow, making them less suitable for the inlet of a mixing valve where consistent directional flow is key.