Check out Fibonacci search and golden section search (the article about Fibonacci search talks about an array, but the technique is really applicable just like golden section search to continuous functions). Fibonacci search is a tiny bit faster. The trick is that you can reuse the points from one iteration to the next. For Fibonacci, you'll have to determine the number of iterations beforehand. No big deal, you know the precision sought anyway.
It can be shown that if you just compare the function values for relative order, Fibonacci search is fastest possible. If you consider the actual values, some form of quasi-Newton is faster.