Create API Gateway APIs in each target Region and use Amazon Route 53 latency-based routing with health checks; integrate each API with a same-Region Lambda function and access a DynamoDB global table is correct because it provides active active Regional endpoints that route customers to the nearest API for consistently low latency while maintaining a single replicated datastore across Regions.

Amazon Route 53 latency based routing directs traffic to the Region with the lowest network latency and health checks keep endpoints available. Keeping API Gateway and Lambda in the same Region reduces cross Region network hops and improves response times. DynamoDB global tables replicate data across Regions so the application has a consistent, highly available dataset without manual synchronization.

Deploy a single Region API Gateway edge-optimized endpoint and integrate with a regional Lambda function; store data in a DynamoDB global table is incorrect because edge optimized endpoints terminate through CloudFront and still route to a single Region for execution, so they do not provide an active active multi Region API deployment for consistently low latency.

Create API Gateway APIs in two Regions and use Amazon Route 53 failover routing with health checks; integrate each API with a Lambda function in the same Region and read and write to a DynamoDB global table is incorrect because failover routing is active passive and does not deliver low latency to all geographies during normal operation. Also deploying in only two Regions does not meet the requirement to run across at least three Regions.

Create API Gateway APIs in each Region and add Route 53 health checks for each record; integrate each API with a local Lambda function and update a DynamoDB table that exists only in that Region is incorrect because per Region DynamoDB tables create data silos and do not provide cross Region replication or consistency. Health checks alone do not steer users to the lowest latency endpoint.

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