I'm looking at this problem, $x^{y}=y^{x}$ and the question is to find the point of intersection of the two curves that form a solution set. I understand how to find the point of intersection using a different method, but I don't understand the following solution. Why is it when "f(x)" is at a minimum or maximum, we have intersection of the two curves?
http://math.berkeley.edu/sites/default/files/pages/solutions_1.pdf