Behavior of Accumulation Functions

By the Second Fundamental Theorem, an accumulation function $g(x) = \int_a^x f(t)\,dt$ of a continuous $f$ has derivative $g' = f$. So the graph of $f$ is the graph of $g'$, and the derivative tests read the behavior of $g$ off it. The figures below show this for $f(t) = \cos t$ on $[0, 2\pi]$, where $g(x) = \int_0^x \cos t\,dt = \sin x$.

On $\left(0, \frac{\pi}{2}\right)$ the graph of $f$ is above the axis, and $g$ increases. At $\frac{\pi}{2}$, $f$ crosses the axis going down, and $g$ has a relative maximum. On $(0, \pi)$, $f$ falls, and $g$ is concave down. At $\pi$, $f$ reaches its lowest point and starts to rise, so $g$ changes to concave up there: a point of inflection.

On a closed interval, an absolute extremum of $g$ occurs at a critical point or at an endpoint. The values of $g$ there are integrals of $f$, so they come from areas.

When $p$ is continuous, the critical points of $G(x) = \int_a^x p(t)\,dt$ are the zeros of $p$. If $p$ is known only at a few points, the Intermediate Value Theorem guarantees zeros: where $p$ has opposite signs at two points, $p = 0$ somewhere between them.