Exam AWS Certified Developer Associate topic 1 question 374 discussion

C is correct. You might wonder why is A wrong: you don't need a private REST API. Private APIs allow requests originating from the same VPC only. Since this is for a public website, you don't need that.

https://docs.aws.amazon.com/apigateway/latest/developerguide/setup-http-integrations.html

B is right

HTTP_PROXY integration would be better, but not on the list.

COST EFFECTIVELY = only API Gateway that can integrate the API Key in the "Method Request" HTTP Header and functions as D but without Lambda

Use Amazon API Gateway to create a REST API. Create an AWS Lambda proxy integration. Make calls to the third-party HTTP API from the Lambda function. Pass the company’s API key as an HTTP request header. This solution uses Amazon API Gateway to create a publicly accessible REST API, but the actual calls to the third-party web service are made from an AWS Lambda function. The company can securely store the API key in the Lambda function's environment variables and pass the key in the headers of the requests made from the Lambda function. This way the key is not exposed to the users of the web application. This solution is cost effective as the cost of Amazon API Gateway is calculated by the number of requests and data transfer. And Lambda is pay as you go service, so you only pay for what you use.

Why not go with D? No need to create private REST Api.