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show f is a contraction mapping

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Poirot

Banned
Feb 15, 2012
250
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

Well-known member
MHB Math Helper
Jan 29, 2012
661
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 $K=\dfrac{1}{3\sqrt{3}}<1$