TITLE: Generalizing the hardest question on the practice math GRE
QUESTION [6 upvotes]: The most-missed question on the Math GRE is the following:
How many times does $x^{12}$ intersect $e^x$?
Because I told you it was hard, you probably realized it was a trick and got the right answer; or you were among the ~25%  who got it right on the test.
I thought this was fun! My question is:
What is the maximum number of times that $e^x$ can intersect a polynomial of degree $n$?

REPLY [2 votes]: You can always get $n+1$ intersections from a polynomial of degree $n$.  Just pick $n+1$ points on $e^x$ and interpolate.  As the GRE question hints, if the leading coefficient is positive, the polynomial crosses from below to above $e^x$ at the highest interpolating point, and $e^x$ will exceed the polynomial one more time.  This gives $n+2$ by taking $n$ points below $0$ and one point above zero, say $4$.  There are no more going toward $-\infty$-you are already on the right side of $e^x$.  The question is whether you can exploit the bend in $e^x$ to make $2,4,\dots $ more intersections by choosing the points correctly.