Interpreting the Behavior of Accumulation Functions Involving Area

g(x) is built by sliding a right endpoint x along the t-axis and totaling signed area under f from some fixed start to x. The fundamental theorem says g'(x) = f(x), so everything you know about using a derivative graph to read off a function's behavior applies directly: g increases while f is positive, decreases while f is negative, and has a critical point wherever f(x) = 0.

To classify that critical point, check which way f crosses zero. If f goes from positive to negative at x = c, g was climbing and starts falling, so x = c is a maximum. If f goes from negative to positive, g was falling and starts climbing, so x = c is a minimum. Concavity of g comes from whether f itself is increasing or decreasing: g is concave up where f is increasing, concave down where f is decreasing, since g''(x) = f'(x). None of this requires ever writing an antiderivative formula for g — the shape of f's graph tells you everything about the shape of g.

This is the qualitative partner to the computational side of unit six. Once you have found where the max or min sits, you still evaluate g at that point by actually integrating, usually with the power rule for antiderivatives. The skill here is matching behavior on the f-graph to behavior on the g-graph before you ever touch numbers, which is exactly how later topics ask you to sketch or describe accumulation functions from a picture of the integrand alone.