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:

if f′(x)>g′(x) for all x>0, and if f(0)=g(0), then f(x)>g(x) for all x>0.

or, substituting ≥ for > produces the theorem

if f′(x)≥g′(x) for all x>0, and if f(0)=g(0), then f(x)≥g(x) for all x≥0.

which can be proved in a similar way

## Contents

* 1 Proof
* 2 Generalizations
* 2.1 Proof
* 3 Application
* 4 References

## Proof

This principle can be proven by considering the function h(x)=f(x)−g(x). If we were to take the derivative we would notice that for x>0,

h′=f′−g′>0.

Also notice that h(0)=0. Combining these observations, we can use the mean value theorem on the interval [0,x] and get

0<h′(x0)=h(x)−h(0)x−0=f(x)−g(x)x.

By assumption, x>0, so multiplying both sides by x gives f(x)−g(x)>0. This implies f(x)>g(x).

## Generalizations

The statement of the racetrack principle can slightly generalized as follows;

if f′(x)>g′(x) for all x>a, and if f(a)=g(a), then f(x)>g(x) for all x>a.

as above, substituting ≥ for > produces the theorem

if f′(x)≥g′(x) for all x>a, and if f(a)=g(a), then f(x)≥g(x) for all x>a.

### Proof

This generalization can be proved from the racetrack principle as follows:

Consider functions f2(x)=f(x+a) and g2(x)=g(x+a). Given that f′(x)>g′(x) for all x>a, and f(a)=g(a),

f2′(x)>g2′(x) for all x>0, and f2(0)=g2(0), which by the proof of the racetrack principle above means f2(x)>g2(x) for all x>0 so f(x)>g(x) for all x>a.

## 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

ex>x

for all real x. This is obvious for x<0 but the racetrack principle is required for x>0. To see how it is used we consider the functions

f(x)=ex

and

g(x)=x+1.

Notice that f(0)=g(0) and that

ex>1

because the exponential function is always increasing (monotonic) so f′(x)>g′(x). Thus by the racetrack principle f(x)>g(x). Thus,

ex>x+1>x

for all x>0.

## References

* Deborah Hughes-Hallet, et al., Calculus.

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