Let $S$, be the set of all functions $f:\left[0,1\right]\to R$, which are continuous on $\left[0,1\right]$, and differentiable on $\left(0,1\right)$. Then for every $f$ in $S$, there exists $c\in \left(0,1\right)$, depending on $f$, such that.

• A

  $\left|f\left(c\right)-f\left(1\right)\right|<\left(1-c\right)\left|f^{\text{'}}\left(c\right)\right|$
• Correct answer
  B

  $\frac{f\left(1\right)-f\left(c\right)}{1-c}=f^{\text{'}}\left(c\right)$
• C

  $\left|f\left(c\right)+f\left(1\right)\right|<\left(1+c\right)\left|f^{\text{'}}\left(c\right)\right|$
• D

  $\left|f\left(c\right)-f\left(1\right)\right|<\left|f^{\text{'}}\left(c\right)\right|$

Hence, the correct answer is option: 'B'.