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Apps on the App Store can't use private API's so I'm assuming none.

Not quite true. Apps aren’t allowed to use private API’s, but some apps still do.

If that's true, I'm very surprised they pass the App Store review considering how stringent the process is.

Couldn't you just statically analyze the binaries to check for calls to barred functions?

No, because you can build a selector string at run-time and call that.