Prove that if two continuous, first order differentiable curves reach a common value in a single point without intersecting, the tangent line of the two curves in that point must be the same.
My argument is posted as an answer: does it constitute an acceptable solution?
Let's suppose we have two functions $f(x)$ and $g(x)$, where $f(x) > g(x) \forall x \neq t $ and $f(x) = g(x)$ in $x = t$.
Then we can show that $f'(t)$ must be equal to $g'(t)$ because, if not, e.g. if $f'(t)>g'(t)$, then we could find two point $t+i$ and $t-i$ where $f(t+i)>g(t+i)$ and $f(t-i) < g(t-i)$, this last one contradicting our hypotesis, and similarly if $f'(t)<g'(t)$ we could find a point $t+i$ where $f(t+i)<g(t+i)$ contradicting again our hypotesis. Then, as $f(t)$ is equal to $g(t)$ and $f'(t)=g'(t)$ the tangent line is the same.
The same reasoning can be applied if $f(x) < g(x) \forall x \neq t $ just inverting the name of the functions.