A particle moves along a straight line and has position at time given by the function . The initial velocity of the particle is and the acceleration function is continuous, given by for all . Prove that for all in some interval where

* Proof. * From the second fundamental theorem of calculus, we have

Then, since implies implies is increasing for all , we have

Now, let and and we have for all where and , as requested

Would this proof still work for a strict inequality (continuous acceleration f”(t) >= 6 for all t in the interval 0 < t < 1) or do we need to include continuity on t = 0 and t = 1?