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 [math]\displaystyle{ f'(x)\gt g'(x) }[/math] for all [math]\displaystyle{ x\gt 0 }[/math], and if [math]\displaystyle{ f(0)=g(0) }[/math], then [math]\displaystyle{ f(x)\gt g(x) }[/math] for all [math]\displaystyle{ x\gt 0 }[/math].

or, substituting ≥ for > produces the theorem

if [math]\displaystyle{ f'(x) \ge g'(x) }[/math] for all [math]\displaystyle{ x\gt 0 }[/math], and if [math]\displaystyle{ f(0)=g(0) }[/math], then [math]\displaystyle{ f(x) \ge g(x) }[/math] for all [math]\displaystyle{ x \ge 0 }[/math].

which can be proved in a similar way

Proof

This principle can be proven by considering the function [math]\displaystyle{ h(x) = f(x) - g(x) }[/math]. If we were to take the derivative we would notice that for [math]\displaystyle{ x\gt 0 }[/math],

[math]\displaystyle{ h'= f'-g'\gt 0. }[/math]

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

[math]\displaystyle{ 0 \lt h'(x_0)= \frac{h(x)-h(0)}{x-0}= \frac{f(x)-g(x)}{x}. }[/math]

By assumption, [math]\displaystyle{ x\gt 0 }[/math], so multiplying both sides by [math]\displaystyle{ x }[/math] gives [math]\displaystyle{ f(x) - g(x) \gt 0 }[/math]. This implies [math]\displaystyle{ f(x) \gt g(x) }[/math].

Generalizations

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

if [math]\displaystyle{ f'(x)\gt g'(x) }[/math] for all [math]\displaystyle{ x\gt a }[/math], and if [math]\displaystyle{ f(a)=g(a) }[/math], then [math]\displaystyle{ f(x)\gt g(x) }[/math] for all [math]\displaystyle{ x\gt a }[/math].

as above, substituting ≥ for > produces the theorem

if [math]\displaystyle{ f'(x) \ge g'(x) }[/math] for all [math]\displaystyle{ x\gt a }[/math], and if [math]\displaystyle{ f(a)=g(a) }[/math], then [math]\displaystyle{ f(x) \ge g(x) }[/math] for all [math]\displaystyle{ x\gt a }[/math].

Proof

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

Consider functions [math]\displaystyle{ f_2(x)=f(x+a) }[/math] and [math]\displaystyle{ g_2(x)=g(x+a) }[/math]. Given that [math]\displaystyle{ f'(x)\gt g'(x) }[/math] for all [math]\displaystyle{ x\gt a }[/math], and [math]\displaystyle{ f(a)=g(a) }[/math],

[math]\displaystyle{ f_2'(x)\gt g_2'(x) }[/math] for all [math]\displaystyle{ x\gt 0 }[/math], and [math]\displaystyle{ f_2(0)=g_2(0) }[/math], which by the proof of the racetrack principle above means [math]\displaystyle{ f_2(x)\gt g_2(x) }[/math] for all [math]\displaystyle{ x\gt 0 }[/math] so [math]\displaystyle{ f(x)\gt g(x) }[/math] for all [math]\displaystyle{ x\gt a }[/math].

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

[math]\displaystyle{ e^{x}\gt x }[/math]

for all real [math]\displaystyle{ x }[/math]. This is obvious for [math]\displaystyle{ x\lt 0 }[/math] but the racetrack principle is required for [math]\displaystyle{ x\gt 0 }[/math]. To see how it is used we consider the functions

[math]\displaystyle{ f(x)=e^{x} }[/math]

and

[math]\displaystyle{ g(x)=x+1. }[/math]

Notice that [math]\displaystyle{ f(0) = g(0) }[/math] and that

[math]\displaystyle{ e^{x}\gt 1 }[/math]

because the exponential function is always increasing (monotonic) so [math]\displaystyle{ f'(x)\gt g'(x) }[/math]. Thus by the racetrack principle [math]\displaystyle{ f(x)\gt g(x) }[/math]. Thus,

[math]\displaystyle{ e^{x}\gt x+1\gt x }[/math]

for all [math]\displaystyle{ x\gt 0 }[/math].

References

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

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