The graph of a function f consists of three straight-line segments that join the points (0,0), (2,3), (4,0), and (6,−2) in the xy-plane. A particular solution F of the differential equation dxdF=f(x) satisfies F(0)=1.
Use the information provided to answer the following.
(a) Calculate the net signed area enclosed by the graph of f and the x-axis on the interval [0,4]. Show all work based on geometric areas.
(b) Determine the exact value of F(4).
(c) For 0≤x≤6, identify the x-value(s) at which F attains a minimum, and justify your conclusion using properties of f. No calculus of F beyond its defining relationship with f is required.