Prove that for every real $x,y \in [1,2]$ is $|x^x - y^y| \leq 4(1+\ln2)|x-y|$

Question

So I have to prove that for every real $x,y \in [1,2]$ is $$|x^x - y^y| \leq 4(1+\ln2)|x-y|$$

I think I have to use Lagrangee but I do not know how to start.

Any help?

Answer 1

It is immediate from MVT: let $f(x)=x^{x}$. then $|f(x)-f(y)| =|x-y| |f'(t)|$ for some $t$ between $x$ and $y$, hence between $1$ and $2$. Note that $f'(t)=\frac d {dt} e^{t\ln t} =(1+\ln t) e^{t\ln t} \leq (1+\ln 2)2^{2}=4(1+\ln 2)$.