Proof by Contradiction

Suppose is differentiable on an interval  with , and  has more than 1 fixed point. Let  such that  are fixed points where . By the definition of fixed point from Weekly HW 6,  and . By Theorem 5.2.3, since  is differentiable on , then  is continuous on . Thus, we can apply the Mean Value Theorem so that there exists a  such that . By algebra, . This is a contradiction because . Thus,  has at most one fixed point.