TITLE: Good final exam bonus problem for calculus students?
QUESTION [8 upvotes]: I'm making a final exam for a first course in calculus at a university.  I need some suggestions for a good bonus problem.  By "good" I mean an interesting problem that some of the brighter students could solve in about 15 minutes.  
Hopefully it should involve limits, derivatives, or integrals. 
The problem should NOT involve trig functions.
The problem should NOT be theoretical. These students can't do real analysis proofs, for instance.

REPLY [2 votes]: "By considering the turning point(s) of the function
$f(x) = x^{1 \over x}$ on $(0, \infty)$ prove that $e^{\pi} > \pi^e$."
Differentiating reveals a global maximum at $x = e$ implying $f(e) > f(\pi)$ and the result follows after raising each side to ${\pi}e$.