Interpreting the Behavior of Accumulation Functions

If g(x) is built by accumulating a rate function f from some starting point, then g' = f. That single fact unlocks everything here. Where f is positive, area is being added, so g is increasing. Where f is negative, area is being subtracted, so g is decreasing. A local max of g happens exactly where f crosses from positive to negative, because that's where g stops gaining and starts losing. A local min happens at the reverse crossing. You never need g's formula to say this: the rate graph tells you everything about the shape of the total.

Concavity works one derivative up. g is concave up where g' = f is increasing, and concave down where f is decreasing. An inflection point of g sits where f changes from increasing to decreasing or vice versa, which is where f' = 0. So reading g off f is like reading position off a velocity graph, and reading g's concavity off f is like reading velocity's behavior off acceleration. This is the same Fundamental Theorem machinery from earlier in the unit, now read as a picture instead of a formula: g(x) = integral from a to x of f(t) dt, and every feature of g traces back to the sign and slope of f.

The one trap is confusing area with steepness. A tall spike in f over a tiny width contributes little area, so g barely moves even though f looks dramatic. What matters for how much g rises is the actual accumulated area, not the height of the curve at any instant.