AP Calculus
posted by cel .
Show that the equation x^3  15x + c = o has exactly one real root.
All I know is that it has something to do with the Mean Value Theorem/Rolle's Theorem.

You must have a typo.
There are all kinds of values of c for which the above equation has 3 roots.
e.g. if c=0
x^3  15x = 0
x(x^2  15) = 0
x = 0 or x = ± √15
are you sure it wasn't x^3 + 15x + c = 0 ?
I will assume it was
It's been 50 years but if I recall, Rolle's theorem says that if there are two xintercepts , then the derivative must be zero between those two points
so let f(x) = x^3 + 15x + c
f '(x) = 3x^2 +15 = 0
notice this is always positive, since x^2 = 5 has no real solution,
so there are no turning points, and the curve cuts only once.
You do know that every cubic must cut he xaxis at least once? 
The missing info it the interval [2,2]. In this interval there could be just one root.