If f is continuous on [a,b], differentiable on (a, b) (imply continuous on [a, b]), and f(a)=f(b), then there exist at least one number c on (a,b) such that f’(c)=0.

Why

• So there is no gap between the function or the differentiated function
• So, if f(a)=f(b), there must be a number c on (a, b) such that f’(c)=0.
  • Since if the function do change (not a horizontal line) , there must point to inverse the change. positive then negative, negative then positive.
  • Since if the function do not change, there is no change of the rate of the function f.
• It is on the (a, b) interval for the reason of the lack of ability to double differentiate.