(Which authentication method allows a web service installed on a network operating system to prove...

One-way server authentication is the standard model used by most TLS-enabled web services to prove the server’s identity to a client. In this model, the server presents an X.509 certificate during the TLS handshake. The client validates the certificate chain to a trusted root CA, checks hostname binding (CN/SAN), validates validity dates, and may check revocation status. If validation succeeds, the client gains cryptographic assurance that it is communicating with the holder of the private key corresponding to the server certificate’s public key, and that the certificate is issued to the expected domain/identity. This proves the server’s identity to the customer without requiring the customer to present a certificate. Mutual authentication would require both client and server to authenticate each other using certificates (commonly in certain enterprise APIs), but the question asks specifically about the web service proving its identity to the customer, which is satisfied by server-only authentication. One-way client authentication is the opposite direction (client proves identity to server). “End-to-end authentication” is a broader concept and not the specific TLS identity proof mechanism described here. Thus, one-way server authentication is the correct choice.