Pick out the true statements:  

a. $|\cos^2 x − \cos^2 y| \le|x − y|$ for all $x, y \in \mathbb{R}$.  

b. If $f : \mathbb{R} \to \mathbb{R}$ satisfies
$|f(x) − f(y)|\le|x − y|^{\sqrt{2}}$for all $x, y \in \mathbb{R}$ then $f$ must be a constant function.  

c. Let$f : \mathbb{R} \to \mathbb{R}$ be continuously differentiable and such that $|f'(x)| \le4/5$ for all $x \in \mathbb{R}$. Then, there exists a unique $x \in \mathbb{R}$ such that $f(x) = x$.

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(a) using Lagranges theorem I get that this is true.  

But I have no idea about the others.Can I get some help please?