Question

I'm quite confused with the racetrack principle and need help on this problem. . . Suppose that f(t) is continuous and twice-differentiable for t>= 0. Further suppose f''(t) >= 3 for all t>= 0 and f(0) = f'(0) = 0.. . Using the Racetrack Principle, what linear function g(t) can we prove is less than f'(t) (for t>= 0)? . g(t) = . . Then, also using the Racetrack Principle, what quadratic function h(t) can we prove is less than than f(t) (for t>= 0)? . h(t) = .

Answer 1

Race track principle says that if two functions are equal at t=0, then the one which has a greater derivative will be greater.

In this case we're comparing f′(t) and g′(t). So we make sure that g(0)=f′(0) and that f′′(t)≥g′(t)

g(t)=at+b

Since it is a line.
g′(t)=a

f′′(t)≥3≥g′(t)⟹3≥a

So let a=3.

f′(0)=0=g(0)=3(0)+b⟹b=0


So that means
g(t)=3t

Do something similar for h(t) starting with
h(t)=at2+bt+c

h(0)=f(0)⟹c=0

So

h(t)=at2+bt