TITLE: Can one find a line that is tangent to a cubic polynomial more than once?
QUESTION [5 upvotes]: I know that any line cannot be tangent to the graph of $y=Ax^3+Bx^2+Cx+D$ at more than one point.
Question: how can one show this, or even prove it?

REPLY [2 votes]: Suppose the cubic has a tangent line with slope $m$, and for the sake of contradiction assume the tangent line touches at 2 distinct points $x=a$ and $x=b$.
By the mean value theorem, there must be a $c$ such that $a < c < b$ and $f'(c) = m$.  This means that $f'(x) = m$ has three solutions $\{a, b, c\}$, but as $f'(x) = m$ is a quadratic, it can only have 2 solutions.