# minimum distance between graphs of functions

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? – Brian M. Scott Sep 19 '12 at 7:32
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. – Najib Idrissi Sep 19 '12 at 7:47
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? – celtschk Sep 19 '12 at 8:10

Your conjecture is false, as @nik shows in a comment. However, consider the function $$\delta \colon x \mapsto (f(x)-g(x))^2,$$ which represents the square of the vertical distance between the graphs of $f$ and $g$. Now, $$\delta'(x)=2 (f(x)-g(x))(f'(x)-g'(x)).$$ If $x_0$ is a minimum of $\delta$, then either $f(x_0)=g(x_0)$, or $f'(x_0)=g'(x_0)$. This suggests that you should modify your conjecture as follows:
Assume $f$ and $g$ are differentiable on $(a,b)$. If the graphs of $f$ and $g$ do not cross and if the infimum of their distance is attained at some $x_0$, then the graphs of $f$ and $g$ are parallel at $x_0$ (in the sense that the tangent lines to the graphs at $x_0$ are parallel).
Be careful. Consider $f(x)=-x$ and $g(x)=(x-1)^2$ on (say) $[0,2]$. If you define the distance as $\inf\{|f(x)-g(y)| \mid x,y \in [0,2]\}$, then the distance between $f$ and $g$ is zero: $f(0)=0=g(1)$. You should decide what is the distance between the graphs of two functions. – Siminore Sep 19 '12 at 8:58
@Geokal: Maybe you should add a more complete explanation what distance you mean. Since the first (and — at least to me, and obviously also to Siminore — most obvious) interpretation doesn't apply, here's my next-best guess: Do you mean the minimal distance of the curves in the plane, that is $\min_{x_1,x_2}\sqrt{(x_1-x_2)^2+(f(x_1)-f(x_2))^2}$? In that case, as long as the curves have well defined tangents in the points realizing the minimum, the answer is even simpler: The connecting line between the points must be orthogonal on both tangents, or else you could make the distance smaller … – celtschk Sep 19 '12 at 12:25