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 ) {\displaystyle f'(x)>g'(x)} for all x > 0 {\displaystyle x>0} , and if f ( 0 ) = g ( 0 ) {\displaystyle f(0)=g(0)} , then f ( x ) > g ( x ) {\displaystyle f(x)>g(x)} for all x > 0 {\displaystyle x>0} . or, substituting ≥ for > produces the theorem
if f ′ ( x ) ≥ g ′ ( x ) {\displaystyle f'(x)\geq g'(x)} for all x > 0 {\displaystyle x>0} , and if f ( 0 ) = g ( 0 ) {\displaystyle f(0)=g(0)} , then f ( x ) ≥ g ( x ) {\displaystyle f(x)\geq g(x)} for all x ≥ 0 {\displaystyle x\geq 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 ) {\displaystyle h(x)=f(x)-g(x)} . If we were to take the derivative we would notice that for x > 0 {\displaystyle x>0} ,
h ′ = f ′ − g ′ > 0. {\displaystyle h'=f'-g'>0.}
Also notice that h ( 0 ) = 0 {\displaystyle h(0)=0} . Combining these observations, we can use the mean value theorem on the interval [ 0 , x ] {\displaystyle [0,x]} and get
0 < h ′ ( x 0 ) = h ( x ) − h ( 0 ) x − 0 = f ( x ) − g ( x ) x . {\displaystyle 0<h'(x_{0})={\frac {h(x)-h(0)}{x-0}}={\frac {f(x)-g(x)}{x}}.}
By assumption, x > 0 {\displaystyle x>0} , so multiplying both sides by x {\displaystyle x} gives f ( x ) − g ( x ) > 0 {\displaystyle f(x)-g(x)>0} . This implies f ( x ) > g ( x ) {\displaystyle f(x)>g(x)} .
Generalizations The statement of the racetrack principle can slightly generalized as follows;
if f ′ ( x ) > g ′ ( x ) {\displaystyle f'(x)>g'(x)} for all x > a {\displaystyle x>a} , and if f ( a ) = g ( a ) {\displaystyle f(a)=g(a)} , then f ( x ) > g ( x ) {\displaystyle f(x)>g(x)} for all x > a {\displaystyle x>a} . as above, substituting ≥ for > produces the theorem
if f ′ ( x ) ≥ g ′ ( x ) {\displaystyle f'(x)\geq g'(x)} for all x > a {\displaystyle x>a} , and if f ( a ) = g ( a ) {\displaystyle f(a)=g(a)} , then f ( x ) ≥ g ( x ) {\displaystyle f(x)\geq g(x)} for all x > a {\displaystyle x>a} .
Proof This generalization can be proved from the racetrack principle as follows: Consider functions f 2 ( x ) = f ( x + a ) {\displaystyle f_{2}(x)=f(x+a)} and g 2 ( x ) = g ( x + a ) {\displaystyle g_{2}(x)=g(x+a)} . Given that f ′ ( x ) > g ′ ( x ) {\displaystyle f'(x)>g'(x)} for all x > a {\displaystyle x>a} , and f ( a ) = g ( a ) {\displaystyle f(a)=g(a)} ,
f 2 ′ ( x ) > g 2 ′ ( x ) {\displaystyle f_{2}'(x)>g_{2}'(x)} for all x > 0 {\displaystyle x>0} , and f 2 ( 0 ) = g 2 ( 0 ) {\displaystyle f_{2}(0)=g_{2}(0)} , which by the proof of the racetrack principle above means f 2 ( x ) > g 2 ( x ) {\displaystyle f_{2}(x)>g_{2}(x)} for all x > 0 {\displaystyle x>0} so f ( x ) > g ( x ) {\displaystyle f(x)>g(x)} for all x > a {\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
e x > x {\displaystyle e^{x}>x}
for all real x {\displaystyle x} . This is obvious for x < 0 {\displaystyle x<0} but the racetrack principle can be used for x > 0 {\displaystyle x>0} . To see how it is used we consider the functions
f ( x ) = e x {\displaystyle f(x)=e^{x}}
and
g ( x ) = x + 1. {\displaystyle g(x)=x+1.}
Notice that f ( 0 ) = g ( 0 ) {\displaystyle f(0)=g(0)} and that
e x > 1 {\displaystyle e^{x}>1}
because the exponential function is always increasing (monotonic) so f ′ ( x ) > g ′ ( x ) {\displaystyle f'(x)>g'(x)} . Thus by the racetrack principle f ( x ) > g ( x ) {\displaystyle f(x)>g(x)} . Thus,
e x > x + 1 > x {\displaystyle e^{x}>x+1>x}
for all x > 0 {\displaystyle x>0} .
References Deborah Hughes-Hallet, et al., Calculus.