# differentiation help

Question from IsaacPhysics:
The displacement x of a damped, simple harmonic oscillator as a function of time t is given by x=Ae^(−αt) cos(ωt+ϕ). Find a general expression for the times at which the displacement is a maximum or minimum. (Your answer will involve an inverse trig function and an arbitrary integer n.)

My (wrong) answer: t = (arctan( - (alpha)/(omega)) - phi)/(omega)
I differentiated the expression for x with respect to t, and set dx/dt = 0 and rearranged for t.
I'm not sure why it's wrong or where to get the 'arbitrary integer n' from?
Looks about right though it would be handy to have the question link to check. The way youve written it is that there is a single solution (corresponding to arctan) whereas obviously there are muliptle solutions corresponding to the "+ n*pi".
Original post by mqb2766
Looks about right though it would be handy to have the question link to check. The way youve written it is that there is a single solution (corresponding to arctan) whereas obviously there are muliptle solutions corresponding to the "+ n*pi".

Thank you so much - got the answer now!
(part D of https://isaacphysics.org/questions/stationary_points4?board=d1084b81-6d7b-448f-a695-b76606b43f37&stage=a_level)