Consider the function f(t)=∣t−1∣ on the interval [0,3]. Let the accumulation function be F(x)=∫0xf(t)dt for 0≤x≤3.
(a) Derive a piecewise formula for F(x) by evaluating the integral directly. Simplify each piece.
(b) Determine each interval on [0,3] where F is differentiable and each interval where F is twice differentiable. Justify carefully.
(c) On the intervals where F′′(x) exists, establish the sign of F′′(x) and hence identify where F is concave up and where it is concave down.
(d) Classify x=1 with respect to differentiability, concavity, and inflection. Provide a concise logical argument for each classification.