increasing-function test
If at every instant your speed is positive, you keep moving forward and never backtrack — so your position increases over time. The increasing-function test is exactly this intuition made rigorous: a function whose slope is always positive on an interval must be rising across that whole interval.
Statement: let f be continuous on an interval I and differentiable on its interior. If f'(x) > 0 for all interior x, then f is strictly increasing on I; if f'(x) >= 0 throughout, then f is non-decreasing (and f'(x) <= 0 gives non-increasing, f'(x) = 0 gives constant). The proof is a direct application of the mean value theorem: for any a < b in I, f(b) - f(a) = f'(c)(b - a) for some c, and the sign of f'(c) controls the sign of f(b) - f(a).
Two honest subtleties. First, strict increase does not require f' > 0 everywhere — f(x) = x^3 is strictly increasing yet f'(0) = 0, so positivity of the derivative is sufficient but not necessary for strict monotonicity. Second, the interval is essential: the test concludes monotonicity on a connected interval, not on a disconnected domain, where f' > 0 on each piece gives no relation between the pieces.
f(x) = x^3 has f'(x) = 3x^2 >= 0, with f'(0) = 0 the only zero; f is nonetheless strictly increasing on all of R — a single point of zero slope does not stop the climb.
Positive derivative forces increase; the converse direction tolerates isolated flat spots.
A clean sufficient condition for strict increase that allows isolated zeros: if f' >= 0 on the interval and f' = 0 only at isolated points (so f' > 0 nowhere on a subinterval), then f is still strictly increasing.