Lagrange's theorem won't help you there.

Let function $f(x)=\dfrac{1}{x}-x + a \ln x$ for $x \in \mathbb R, x>0$ , where $a$ is a parameter and $a \in \mathbb R$ .

If there exists $x_1,x_2 \in (0,\infty), x_1>x_2$ and $f'(x_1)=f'(x_2)=0$ , is it always true that $\dfrac{f(x_1)-f(x_2)}{x_1-x_2}<a-2$ ?