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