Which of the following statements best explains why
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Which of the following statements best explains why, when determining the absolute extrema of a continuous piecewise–defined function on a closed interval, the function must be evaluated at each point where its algebraic formula changes?
(a) Because every point where the formula changes is a critical point where f′(x)=0, so each must be tested.
(b) Because a piecewise function cannot remain continuous at its formula breaks, making those points natural locations for extrema.
(c) Because an absolute extremum can occur at a point where the function’s definition changes even if the derivative does not exist there.
(d) Because the Mean Value Theorem guarantees the existence of extrema at points where the rule for the function changes.