Racetrack principle

In calculus,
the racetrack principle describes the movement and growth of two functions in terms of their derivatives.
This principle is derived from the fact that if a horse named Frank Fleetfeet always runs faster than a horse named Greg Gooseleg, then if Frank and Greg start a race from the same place and the same time, then Frank will win. More briefly, the horse that starts fast and stays fast wins.
In symbols:
or, substituting ≥ for > produces the theorem
which can be proved in a similar way

Proof

This principle can be proven by considering the function. If we were to take the derivative we would notice that for,
Also notice that. Combining these observations, we can use the mean value theorem on the interval and get
By assumption,, so multiplying both sides by gives. This implies.

Generalizations

The statement of the racetrack principle can slightly generalized as follows;
as above, substituting ≥ for > produces the theorem

Proof

This generalization can be proved from the racetrack principle as follows:
Consider functions and.
Given that for all, and,
for all, and, which by the proof of the racetrack principle above means for all so for all.

Application

The racetrack principle can be used to prove a lemma necessary to show that the exponential function grows faster than any power function. The lemma required is that
for all real. This is obvious for but the racetrack principle can be used for. To see how it is used we consider the functions
and
Notice that and that
because the exponential function is always increasing so. Thus by the racetrack principle. Thus,
for all.