Brain Teaser  Minimum distance between two curves

01122016, 09:40 AM
(This post was last modified: 01122016 12:34 PM by Pekis.)
Post: #15




RE: Brain Teaser  Minimum distance between two curves
Thanks for your refreshing brain teaser.
f(x)=x^{2}+1 => f'(x)=2x g(x)=sqrt(x) => g'(x)=1/(2sqrt(x)) The normal equation on f at x_{f}: Slope: 1/f'(x_{f}) Intercept: f(x_{f})+x_{f}/f'(x_{f}) > y=x/(2x_{f}))+x_{f}^{2}+3/2 The normal equation on g at x_{g}: Slope: 1/g'(x_{g}) Intercept: g(x_{g})+x_{g}/g'(x_{g}) > y=2sqrt(x_{g})x+(2x_{g}+1)sqrt(x_{g}) Same normal => 1/(2x_{f}) must be equal to 2sqrt(x_{g}) and x_{f}^{2}+3/2 must be equal (2x_{g}+1)sqrt(x_{g}) Leads to x_{f}=1/(4sqrt(x_{g})) (or x_{g}=1/(16x_{f}^{2}) and then 4x_{f}^{5}+6x_{f}^{3}x_{f}^{2}1/8=0 Only one real root: approx. 0.331695 for x_{f} => approx. 0.56807 for x_{g} => distance (x_{f},f(x_{f})) (x_{g},g(x_{g})) is approx. 0.42759 

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