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)\geq g'(x)

for all

x>0

, and if

f(0)=g(0)

, then

f(x)\geq g(x)

for all

x\geq 0

.

which can be proved in a similar way

Proof

This principle can be proven by considering the function $h(x)=f(x)-g(x)$ . 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)\geq g'(x)

for all

x>a

, and if

f(a)=g(a)

, then

f(x)\geq 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 is required 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.

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