We need to understand this problem carefully, because when I was first reading it, it seemed as if it said "there is at least one point which is a zero".

We want to show that there isat mostone point. We will argue by contradiction. Say there are more than two points on . Pick any two distinct ones with . Define the function . Note that . Consider this function on the closed interval . We notice that , the function is continous on , and differenciable on . By Rolle's theorem we can find so that . Meaning . But that is impossible because since . Hence a contradiction. That means there can be at most one zero.

Q.E.D.