The area of a plane region depends smoothly on a real parameter k and is represented…
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The area of a plane region depends smoothly on a real parameter k and is represented by a differentiable function A(k). Which of the following best explains why solving A'(k)=0 is used to locate the value of k that yields a maximum or minimum area, provided k lies in the interior of the allowed interval?
(a) Because when A′(k)=0 the graph of A has a horizontal tangent, so interior critical points are candidates for extreme values by the First Derivative Test.
(b) Because whenever A′(k)=0, A must achieve either a maximum or a minimum there, so no other checks are necessary.
(c) Because A′(k)=0 marks the value of k where the area is changing the fastest, thereby revealing its largest or smallest value.
(d) Because critical points occur only where A′(k)=0, so endpoints do not have to be considered when searching for extrema.