Let’s use LMVT for $x\in \left[a,c\right]$

$\frac{f\left(c\right)-f\left(a\right)}{c-a}=f^{\text{'}}\left(\alpha \right),\alpha \in \left(a,c\right)$

Also use LMVT for $x\in \left[c,b\right]$

$\frac{f\left(b\right)-f\left(c\right)}{b-c}=f^{\text{'}}\left(\beta \right),\beta \in \left(c,b\right)$

$\because f^{\text{'}\text{'}}\left(x\right)<0\Rightarrow f^{\text{'}}\left(x\right)$ is decreasing

$f^{\text{'}}\left(\alpha \right)>f^{\text{'}}\left(\beta \right)$

$\frac{f\left(c\right)-f\left(a\right)}{c-a}>\frac{f\left(b\right)-f\left(c\right)}{b-c}$

$\frac{f\left(c\right)-f\left(a\right)}{f\left(b\right)-f\left(c\right)}>\frac{c-a}{b-c}$ ( $\because f\left(x\right)$ is increasing)