The position of the particle moving along the x-axis is given by the function f(t) = 4t^3 - 39t^2 +108t. During which time interval(s) is the particle's speed increasing?

Question

a. t > 3.25

b. t > 4.5

c. 0 < t < 3.25

d. 0 < t < 2 and t > 4.5

e. 2 < t < 3.25 and t > 4.5

Answer 1

Speed will be increasing any time both velocity and acceleration have the same sign:

v(t) = f^'(t) = 12t² - 78t + 108 = 0

2t² - 13t + 18 = 0

(2t - 9)(t - 2) = 0

v(t) > 0 for t < 2 or t > 4.5

a(t) = v^'(t) = f^"(t) = 24t - 78 = 0

4t - 13 = 0

t = 13/4

a(t) > 0 for t > 3.25

Speed is increasing when 2 < t < 3.25 or t > 4.5