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{{No footnotes|date=April 2021}}
In [[calculus]],
the '''racetrack principle''' describes the movement and growth of two functions in terms of their [[derivative]]s.

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>f'(x)>g'(x)</math> for all <math>x>0</math>, and if <math>f(0)=g(0)</math>, then <math>f(x)>g(x)</math> for all <math>x>0</math>.
or, substituting ≥ for > produces the theorem
:if <math>f'(x) \ge g'(x)</math> for all <math>x>0</math>, and if <math>f(0)=g(0)</math>, then <math>f(x) \ge g(x)</math> for all <math>x \ge 0</math>.
which can be proved in a similar way

==Proof==
This principle can be proven by considering the function <math>h(x) = f(x) - g(x)</math>. If we were to take the derivative we would notice that for <math>x>0</math>,

:<math> h'= f'-g'>0.</math>

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

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

By assumption, <math>x>0</math>, so multiplying both sides by <math>x</math> gives <math>f(x) - g(x) > 0</math>. This implies <math>f(x) > g(x)</math>.

==Generalizations==

The statement of the racetrack principle can slightly generalized as follows;
:if <math>f'(x)>g'(x)</math> for all <math>x>a</math>, and if <math>f(a)=g(a)</math>, then <math>f(x)>g(x)</math> for all <math>x>a</math>.

as above, substituting ≥ for > produces the theorem
:if <math>f'(x) \ge g'(x)</math> for all <math>x>a</math>, and if <math>f(a)=g(a)</math>, then <math>f(x) \ge g(x)</math> for all <math>x>a</math>.

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

Consider functions <math>f_2(x)=f(x+a)</math> and <math>g_2(x)=g(x+a)</math>.
Given that <math>f'(x)>g'(x)</math> for all <math>x>a</math>, and <math>f(a)=g(a)</math>,

<math>f_2'(x)>g_2'(x)</math> for all <math>x>0</math>, and <math>f_2(0)=g_2(0)</math>, which by the proof of the racetrack principle above means <math>f_2(x)>g_2(x)</math> for all <math>x>0</math> so <math>f(x)>g(x)</math> for all <math>x>a</math>.

==Application==
The racetrack principle can be used to prove a [[lemma (mathematics)|lemma]] necessary to show that the [[exponential function]] grows faster than any power function. The lemma required is that
:<math> e^{x}>x </math>
for all real <math>x</math>. This is obvious for <math>x<0</math> but the racetrack principle can be used for <math>x>0</math>. To see how it is used we consider the functions
:<math> f(x)=e^{x}</math>
and
:<math> g(x)=x+1.</math>
Notice that <math>f(0) = g(0)</math> and that
:<math> e^{x}>1</math>
because the exponential function is always increasing ([[monotonic]]) so <math>f'(x)>g'(x)</math>. Thus by the racetrack principle <math>f(x)>g(x)</math>. Thus,
:<math> e^{x}>x+1>x</math>
for all <math>x>0</math>.

==References==
* Deborah Hughes-Hallet, et al., ''Calculus''.

[[Category:Differential calculus]]
[[Category:Mathematical principles]]

Racetrack principle - Revision history

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