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

h'(x_{0})={\frac {h(x)-h(0)}{x-0}}={\frac {f(x)-g(x)}{x}}>0.

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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