The follow theorem is called the racetrack principle and is a consequence of the
ID: 3285894 • Letter: T
Question
The follow theorem is called the racetrack principle and is a consequence of the mean value theorem.
If f(x) and g(x) are continuous on [a; b] and dieren-
tiable on (a; b) and f0(x) =< g0(x) on (a; b) then
1. If f(a) = g(a) then f(x) =< g(x) on [a; b].
2. If f(b) = g(b) then f(x) >= g(x) on [a; b].
It is called the racetrack principle because if f and g are racers running continuous and dierentiable paths and f is
running slower than g then
1. If f and g start at the same place then g was ahead of f for the whole race.
2. If f and g end at the same place then g was behind f for the whole race.
Use the racetrack principle to show that sin(x) =< x on [0; t] for any t >=0.