[php] How do I get the last inserted ID of a MySQL table in PHP?

What you wrote would get you the greatest id assuming they were unique and auto-incremented that would be fine assuming you are okay with inviting concurrency issues.
Since you're using MySQL as your database, there is the specific function LAST_INSERT_ID() which only works on the current connection that did the insert.
PHP offers a specific function for that too called mysql_insert_id.