This is starting out as just a list of articles to review. Stay tuned for actual examples.
Beyond increasing MaxClients and MaximumRequestsPerChild, a key recommendation by all these sources is to set KeepAlive to On but use a really low value like 3 or even 2 for the KeepAliveTimeout.
Another suggestion is to co-locate a nginx reverse proxy on the same server as Apache and proxy all traffic nginx to Apache. This is reminiscent of the old webmaster’s trick of putting squid in front of Apache as a caching proxy server.
Nginx configuration examples can be found in the Nginx Wiki.
Of course in a LAMP stack the web server is just one part of the puzzle. Both php and the database server will also need tuning.