Plz help me continuous about Mean Value Theorem

Question

$|\sinh x|\geq|x|, \forall x \in \mathbb{R}$

$$ f(x) = \sinh x $$

$$f'(c)= \frac{f(b)-f(a)}{b-a} $$

$$\cosh (c)= \frac{\sinh b-\sinh a}{b-a} $$

what can I do this continuous?

Answer 1

From $$\cosh c=\frac{\sinh b-\sinh a}{b-a}$$

Let $b=x$ and $a=0$, then we have $$\cosh c=\frac{\sinh x-\sinh 0}{x-0}$$

That is $$|\frac{\sinh x}{x}| = |\cosh c| \ge 1$$

Cross multiply to get $$|\sinh x|\ge |x|$$