Possible duplicates
Is there a reason to not use Boost?
What are the advantages of using the C++ BOOST libraries?
Here are two suggestions for advocating boost:
Who's Using Boost? (http://www.boost.org/users/uses.html)
lots of major projects use boost: (e.g., adobe photoshop, CERN)
Boost Project Cost (http://www.boost.org/development/index.html)
How much it would cost to hire a team to write boost from scratch? There's a nifty (somewhat gimmicky) calculator there that helps to put it in perspective.