$\text{ Let }f\left(x\right)\text{ be any function continuous and twice differentiable function on }(a,b)\text{ . If for }$

$\begin{array}{l}\text{ all }x\in (a,b),f^{\text{'}}\left(x\right)>0\text{ and }{f}^{\text{'}\text{'}}\left(x\right)<0,\text{ then for any }c\in (a,b),\frac{f\left(c\right)-f\left(a\right)}{f\left(b\right)-f\left(c\right)}\text{ is greater }\\ \text{ than }\end{array}$

$\text{ Since }{f}^{\text{'}}\left(x\right)>0,\text{ the function }f\text{ is increasing and since }f"\left(x\right)<0,\text{ the function }$downward facing

$\begin{array}{l}\mathrm{Nm}\\ \text{ So, slope of }\mathrm{AC}\text{ will be greater than slope of }\mathrm{BC}\\ \text{ i.e., }\frac{f\left(c\right)-f\left(a\right)}{c-a}>\frac{f\left(b\right)-f\left(c\right)}{b-c}\Rightarrow \frac{f\left(c\right)-f\left(a\right)}{f\left(b\right)-f\left(c\right)}>\frac{c-a}{b-c}\end{array}$