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 $f^{\prime }(x)>g^{\prime }(x)$ 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 $f^{\prime }(x)\ge g^{\prime }(x)$ for all $x>0$, and if $f(0)=g(0)$, then $f(x)\ge g(x)$ for all $x\ge 0$.

which can be proved in a similar way

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 $x>0$,

$h^{\prime }=f^{\prime }-g^{\prime }>0.$

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

$0<h^{\prime }(x_0)=\frac{h(x)-h(0)}{x-0}=\frac{f(x)-g(x)}{x}.$

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 $f^{\prime }(x)>g^{\prime }(x)$ 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 $f^{\prime }(x)\ge g^{\prime }(x)$ for all $x>a$, and if $f(a)=g(a)$, then $f(x)\ge g(x)$ for all $x>a$.

Proof

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

Consider functions $f_2(x)=f(x+a)$ and $g_2(x)=g(x+a)$. Given that $f^{\prime }(x)>g^{\prime }(x)$ for all $x>a$, and $f(a)=g(a)$,

${f_2}^{\prime }(x)>{g_2}^{\prime }(x)$ for all $x>0$, and $f_2(0)=g_2(0)$, which by the proof of the racetrack principle above means $f_2(x)>g_2(x)$ 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

$e^x>x$

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

$f(x)=e^x$

and

$g(x)=x+1.$

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

$e^x>1$

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

$e^x>x+1>x$

for all $x>0$.

References

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