Racetrack principle

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 $f'(x) \ge g'(x)$ 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 $x>0$ ,

: $h'= f'-g'>0.$

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

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

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 $f'(x)>g'(x)$ 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 $f'(x) \ge g'(x)$ 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 $f'(x)>g'(x)$ for all $x>a$ , and $f(a)=g(a)$ ,

$f_2'(x)>g_2'(x)$ 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 $f'(x)>g'(x)$ . 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.

Category:Differential calculus

Category:Mathematical principles