This article is from the Puzzles FAQ, by Chris Cole firstname.lastname@example.org and Matthew Daly email@example.com with numerous contributions by others.
What is the longest time that a particle can take in travelling between two
points if it never increases its acceleration along the way and reaches the
second point with speed V?
1. x(0) = 0; x(T) = X
2. v(0) = 0; v(T) = V
3. d(a)/dt <= 0
a(t) = constant = A = V^2/2X which implies T = 2X/V.
Consider assumptions as they apply to f(t) = A * t - v(t):
1. integral from 0 to T of f = 0
2. f(0) = f(T) = 0
3. d^2(f)/dt^2 <= 0
From the mean value theorem, f(t) = 0. QED.