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Prove this : When the graphs of two differentiable functions have the minimum distance then the secants at those points are parallel .

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    Do you mean that the *tangent lines* are parallel?2012-09-19
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    Take $f(x)=0$, $g(x)=x$, then the minimum distance between the two graphs is 0, at the point $x=0$, but at this point, neither the secant (I'm guessing orthogonal) lines nor the tangent lines are parallel.2012-09-19
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    Assuming the two curves don't intersect, here's a hint: If the lines are parallel, then what does it tell you about their slopes?2012-09-19

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