Which of the following best explains why a point where the derivative of a continuous…
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
Which of the following best explains why a point where the derivative of a continuous piecewise function fails to exist can still correspond to a relative extremum?
(a) Because continuity always fails at points where the derivative does not exist, forcing an extremum at that point
(b) Because the Mean Value Theorem guarantees there is some c with f′(c)=0 between any two points, so the derivative need not exist at the extremum itself
(c) Because such a point may be a sharp corner, allowing the function to reach a highest or lowest nearby value even though f′ is undefined there
(d) Because an undefined derivative means the tangent is vertical, so the graph stops increasing or decreasing and must attain an extremum