Brain Teaser - Minimum distance between two curves
01-12-2016, 09:40 AM (This post was last modified: 01-12-2016 12:34 PM by Pekis.)
Post: #15
 Pekis Member Posts: 159 Joined: Aug 2014
RE: Brain Teaser - Minimum distance between two curves
Thanks for your refreshing brain teaser.

f(x)=x2+1 => f'(x)=2x
g(x)=sqrt(x) => g'(x)=1/(2sqrt(x))

The normal equation on f at xf: Slope: -1/f'(xf) Intercept: f(xf)+xf/f'(xf)
-> y=-x/(2xf))+xf2+3/2

The normal equation on g at xg: Slope: -1/g'(xg) Intercept: g(xg)+xg/g'(xg)
-> y=-2sqrt(xg)x+(2xg+1)sqrt(xg)

Same normal => -1/(2xf) must be equal to -2sqrt(xg) and xf2+3/2 must be equal (2xg+1)sqrt(xg)

xf=1/(4sqrt(xg)) (or xg=1/(16xf2)
and then
4xf5+6xf3-xf2-1/8=0

Only one real root: approx. 0.331695 for xf => approx. 0.56807 for xg

=> distance (xf,f(xf)) (xg,g(xg)) is approx. 0.42759
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 Messages In This Thread Brain Teaser - Minimum distance between two curves - CR Haeger - 11-18-2015, 04:35 PM RE: Brain Teaser - Minimum distance between two curves - Vtile - 11-18-2015, 05:50 PM RE: Brain Teaser - Minimum distance between two curves - Gerald H - 11-18-2015, 06:18 PM RE: Brain Teaser - Minimum distance between two curves - Vtile - 11-18-2015, 06:24 PM RE: Brain Teaser - Minimum distance between two curves - Jonathan Cameron - 11-18-2015, 06:40 PM RE: Brain Teaser - Minimum distance between two curves - CR Haeger - 11-18-2015, 07:06 PM RE: Brain Teaser - Minimum distance between two curves - Jonathan Cameron - 11-18-2015, 11:47 PM RE: Brain Teaser - Minimum distance between two curves - Vtile - 11-18-2015, 07:23 PM RE: Brain Teaser - Minimum distance between two curves - Werner - 11-18-2015, 10:19 PM RE: Brain Teaser - Minimum distance between two curves - CR Haeger - 11-19-2015, 12:30 AM RE: Brain Teaser - Minimum distance between two curves - Thomas Klemm - 11-21-2015, 01:46 AM RE: Brain Teaser - Minimum distance between two curves - Jonathan Cameron - 11-21-2015, 05:37 PM RE: Brain Teaser - Minimum distance between two curves - Bunuel66 - 11-21-2015, 09:39 PM RE: Brain Teaser - Minimum distance between two curves - quinyu - 01-02-2016, 06:14 PM RE: Brain Teaser - Minimum distance between two curves - Pekis - 01-12-2016 09:40 AM

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