Math Genius - Rolle's Theorem lyrics

If a real-valued function f is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then such a c exists in the open interval (a, b) that f'(c) = 0 Hp. f(x) continuous on [a,b] f(x) is differentiable on (a,b) f(a) = f(b) Th. f'(c) = 0 Graphic proof: Since the function is continuous in [a,b] the arc it draws on the graphic must raise from a, reach a maximum in a certain point c and then reach the x axis again in b. It's easy to see that the slope of the tangent in c must be 0 since the line is horizontal. Mathematical proof: Let's study the increment of the function in a point h. f(c + h) - f(c) = f(c + h) - f(c) >= 0 h > 0 Apply the Theorem of sign permanence lim h->0- [f(c + h) - f(c)]/h lim h->0+ [f(c + h) - f(c)]/h >= 0 h > 0 Use the definition of differentiability lim h- > 0 [f(c + h) - f(c)]/h = 0 Use the definition of derivative f'(c) = 0 Q.E.D.