A particle moves so that its position is given by ⟨cos(t),sin(t),cos(6t)⟩. Find the maximum and minimum speeds of the particle.

Answer 1

r(t) = ⟨cos(t),sin(t),cos(6t)⟩
v = <-sin(t),cos(t),-6sin(6t)>
speed is |v|, so
s^2 = 1+36sin^2(6t)

max speed occurs when ds/dt = 0
2s ds/dt = 216sin^4(6t)
ds/dt=0 when sin(6t) = 0

I expect you can take it from there, no?