show f is a contraction mapping

[Poirot]

f:[1,infinity)->[1,infinity)

$f(x)=x^{0.5}+x^{-0.5}$

I thought about using MVT but it doesn't work and I've tried showing it conventially but i can't reduce it to k|x-y|

 

[Fernando Revilla]

f:[1,infinity)->[1,infinity) $f(x)=x^{0.5}+x^{-0.5}$ I thought about using MVT but it doesn't work and I've tried showing it conventially but i can't reduce it to k|x-y|

For $1\leq x <y<+\infty$ you'll get

$\left|f(y)-f(x)\right|=\dfrac{1}{2\sqrt{c}}\left(1-\dfrac{1}{c}\right)|y-x|$

Now, use that a global maximum for $F(c)=\dfrac{1}{2\sqrt{c}}\left(1-\dfrac{1}{c}\right)$ in $[1,+\infty)$ is $K=\dfrac{1}{3\sqrt{3}}<1$