1. Rolle's Theorem
   If the function is continuous on the closed interval and differentiable on the open interval , and if , then there exists at least one in the open interval such that
   
   
   
   
   If the function is not differentiable on the open interval , then Rolle's Theorem does not hold. For example, the function is continuous on the closed interval and satisfies , but there is no in such that .
   
   
   
   
   
   
2. Mean Value Theorem
   If the function is continuous on the closed interval and differentiable on the open interval , then there exists at least one in the open interval such that
   In other words, if , the Mean Value Theorem reduces to Rolle's Theorem.
   
   
   
   
   
   
3. Applications of the Mean Value Theorem
   If two functions and are continuous on the closed interval and differentiable on the open interval , the following properties hold for all in :
1. If , then is a constant function on the interval .
2. If , then on the interval , where is a constant.