The "peak response" refers to the farthest distance that the body travels (from equilibrium) at any point in its motion. So, generally thinking about these kinds of problems requires thinking about what the motion looks like (by graphing position as a function of time) and deciding which point is farthest from the equilibrium line. If you were to solve for this quantity using a system of equations involving the velocity and position functions then you would solve the velocity equation for time when the velocity itself is zero, then plug that value (in this case values) into the position equation. It is still left to you to decide witch time is the correct one to plug in to the position equation; this will always require a plot of the position vs time.
edit: for a plot of the position as a function of time click on the little radio button above that says "childs-3-12_plot.png", it is much easier to see where the peak response is when graphed.

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