Application of Lagrange theorem with absolute value

Question

I have to solve an exercise using the Lagrange theorem but I have doubt. From the theorem I know that, let a continuous function in $[a,b]$ and differentiable in $(a,b)$ then: $$\exists c\in(a,b): ,\, f'(c)=\frac{f(b)-f(a)}{b-a}\,\,\,\,\, (*)$$ Now I need the equality with the absolute value. Can I say that if (*) holds then it holds also the following? $$|f'(c)|=\left|\frac{f(b)-f(a)}{b-a}\right|$$

Answer 1

For all $x,y \in \mathbb{R}$, $x = y$ implies $|x|=|y|$.