Let f be any function continuous on [a, b] and twice differentiable...

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

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

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

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

$∵ {f}^{''}\left(x\right) < 0 ⇒ {f}^{'}\left(x\right)$ is decreasing

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

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

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