After finding the critical points of the function, how do you find the global maximum and minimum. Is the most efficient way to simply find the value of the function at each of the critical points and the end points. This seems a little tedious. Is there a better way?
For example, say I know that on the interval (0,10), the critical points of f occur at x=1,5,6,8. To find the global maximum on [0,10], do I have to see which of f(0), f(1), f(5), f(6), f(8), and f(10) is the largest? There has to be a better way, isn't there?
Unfortunately no, as far as I know there is no better way than the method you described. You might be able to find a method to find a global minimum without evaluating all the critical points that will work with one particular function or class of functions, but there is no method known that will work for any function . Don't let that stop you though - maybe you can invent one!
To be honest, the problem of finding a global minimum for a given function (one which may have thousands or even millions of variables) using a computer is a difficult problem that is the subject of intense current research. Finding minimums and maximums is a field called optimization and it has huge economic and scientific importance. However, most algorithms are only able to find local minimums and maximums because they cannot evaluate all the points of the function. Global optimization is a very difficult problem.