The velocity of the object at time t is given by vel[t] = 3cos[4t]-4sin[3t] units/sec . the speed of the object is given by speed[t] = abs[3 cos[4 t] - 4 sin[3 t]]units/sec . so speed[t]^2 = vel[t]^2 and speed[t] is greatest when vel[t]^2 is as big as possible, and speed[t] is least when vel[t]^2 is as small as possible. report on the greatest and smallest speed at which this object moves. tip: it's a real mess to try to take the derivative of speed[t] = abs[3 cos[4 t] - 4 sin[3 t]] , but it's not much trouble to take the derivative of speed[t]^2 = (3 cos[4 t] - 4 sin[3 t])^2 . to estimate when the speed is smallest and greatest, estimate when speed[t]^2 = (3 cos[4 t] - 4 sin[3 t])^2 is smallest and greatest