About the Converse of the Mean Value Theorem.




Dr. Wei-Chi Yang

Radford University

Radford, VA 24142

www.radford.edu/~wyang

Theorem

If $f$ is continuous on $[a,b]$ and differentiable on $(a,b),$then there is a point $p\in (a,b)$ such that MATH

Conversely, we can ask under what condition, for all $x\in (a,b)$ there is a $p\in (a,b)$ such that MATHorMATH

Theorem

Suppose f$^{\text{ '}}$ is strictly monotone in the interval $[a,b].$ Then there exists $x_{0}\in (a,b)$ such that

  1. When $x\in (a,x_{0}),$ there exists a unique $p\in (a,b)$ satisfying MATH

  2. When $x\in (x_{0},b),$ there exists a unique $p\in (a,b)$ satisfyingMATHRemark: For CP, click here.