The velocity v (in meters per second) of a particle moving along a horizontal line is described by the following verbal graph for 0≤t≤10 seconds.
• v starts at 4 m/s when t=0, decreases linearly to 0 m/s at t=3, then continues decreasing to −2 m/s at t=4.
• From t=4 to t=6, v increases smoothly and concavely upward to 3 m/s at t=6.
• From t=6 to t=10, v decreases linearly to 1 m/s at t=10.
Throughout the motion, v is differentiable except at the corner points t=3 and t=6, where v fails to be differentiable but is continuous.
(a) Identify the open intervals on which the particle’s position function s is concave up and the intervals on which s is concave down. Explain how the concavity of s is related to the behavior of v.
(b) Determine all times in 0<t<10 at which the particle could attain a local maximum position. Provide a justification that refers to the sign of v just before and just after each candidate time.
(c) Sketch a possible graph of the acceleration a(t) on the axes provided below, labeling all important features. Describe verbally how you determined the sign and relative magnitude of a(t) on each interval.