2025 AP Calc AB & BC Free Response Question #4

This FRQ from the 2025 AP Calculus AB and BC set presents us with a graph consisting of a line segment and two semicircles. Over the course of the problem, we use the Fundamental Theorem of Calculus to determine the derivative of a function defined as a definite integral, utilize the graph of a derivative to determine the location of inflection points, use signed area arguments to determine several values of a function defined as a definite integral, and determine the absolute minimum of the function defined as a definite integral using the Extreme Value Theorem.