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 ${\displaystyle f'(x)>g'(x)}$ for all ${\displaystyle x>0}$, and if ${\displaystyle f(0)=g(0)}$, then ${\displaystyle f(x)>g(x)}$ for all ${\displaystyle x>0}$.

or, substituting ≥ for > produces the theorem

if ${\displaystyle f'(x)\geq g'(x)}$ for all ${\displaystyle x>0}$, and if ${\displaystyle f(0)=g(0)}$, then ${\displaystyle f(x)\geq g(x)}$ for all ${\displaystyle x\geq 0}$.

which can be proved in a similar way

Proof

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

${\displaystyle h'=f'-g'>0.}$

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

${\displaystyle 0<h'(x_{0})={\frac {h(x)-h(0)}{x-0}}={\frac {f(x)-g(x)}{x}}.}$

By assumption, ${\displaystyle x>0}$, so multiplying both sides by ${\displaystyle x}$ gives ${\displaystyle f(x)-g(x)>0}$. This implies ${\displaystyle f(x)>g(x)}$.

Generalizations

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

if ${\displaystyle f'(x)>g'(x)}$ for all ${\displaystyle x>a}$, and if ${\displaystyle f(a)=g(a)}$, then ${\displaystyle f(x)>g(x)}$ for all ${\displaystyle x>a}$.

as above, substituting ≥ for > produces the theorem

if ${\displaystyle f'(x)\geq g'(x)}$ for all ${\displaystyle x>a}$, and if ${\displaystyle f(a)=g(a)}$, then ${\displaystyle f(x)\geq g(x)}$ for all ${\displaystyle x>a}$.

Proof

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

Consider functions ${\displaystyle f_{2}(x)=f(x+a)}$ and ${\displaystyle g_{2}(x)=g(x+a)}$. Given that ${\displaystyle f'(x)>g'(x)}$ for all ${\displaystyle x>a}$, and ${\displaystyle f(a)=g(a)}$,

${\displaystyle f_{2}'(x)>g_{2}'(x)}$ for all ${\displaystyle x>0}$, and ${\displaystyle f_{2}(0)=g_{2}(0)}$, which by the proof of the racetrack principle above means ${\displaystyle f_{2}(x)>g_{2}(x)}$ for all ${\displaystyle x>0}$ so ${\displaystyle f(x)>g(x)}$ for all ${\displaystyle x>a}$.

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

${\displaystyle e^{x}>x}$

for all real ${\displaystyle x}$. This is obvious for ${\displaystyle x<0}$ but the racetrack principle can be used for ${\displaystyle x>0}$. To see how it is used we consider the functions

${\displaystyle f(x)=e^{x}}$

and

${\displaystyle g(x)=x+1.}$

Notice that ${\displaystyle f(0)=g(0)}$ and that

${\displaystyle e^{x}>1}$

because the exponential function is always increasing (monotonic) so ${\displaystyle f'(x)>g'(x)}$. Thus by the racetrack principle ${\displaystyle f(x)>g(x)}$. Thus,

${\displaystyle e^{x}>x+1>x}$

for all ${\displaystyle x>0}$.

References

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