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Given that two real valued functions $f(x)$ and $g(x)$ have one unique solution $c$ such that $f(c)=g(c)$, why must the slope of the tangent lines of the curves at that point be equal?

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  • $\begingroup$ That's not true as stated. If $f(x) = x$ and $g(x) = -x$, then $f(0)=g(0) = 0$ is the only intersection, but $f'(0) = 1 \ne -1 = g'(0)$. $\endgroup$ – Alexis Olson Oct 5 '16 at 1:34
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$f(x)=x$ and $g(x)=1$ intersect at $x=1$ exclusively, where $f'(1)=1$ and $g'(1)=0$.

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  • $\begingroup$ Fair.What is the answerer talking about in this post then:math.stackexchange.com/questions/1426353/… $\endgroup$ – Drew Meier Oct 5 '16 at 1:37
  • $\begingroup$ @DrewMeier This is rather specific to the functions being discussed. For those functions there are $0$, $1$, or $2$ solutions. For these particular functions, meeting at exactly $1$ point implies they are tangent at that point. $\endgroup$ – Alexis Olson Oct 5 '16 at 2:05
  • $\begingroup$ Ah that makes sense, thanks! $\endgroup$ – Drew Meier Oct 7 '16 at 15:07

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