A cost function is defined and differentiable on the closed interval
MATHEMATICS · Single correct · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives
Problem
A cost function C(x) is defined and differentiable on the closed interval 0≤x≤12. A student finds that the only critical point of C in (0,12) is at x=5 and immediately concludes that C attains its minimum there. Which of the following best explains why the student’s conclusion may be incomplete?
(a) A function’s global minimum on a closed interval must be compared with its values at the endpoints as well as at critical points.
(b) Because the second derivative at x=5 was not examined, x=5 could actually correspond to a maximum rather than a minimum.
(c) The derivative test is invalid on a closed interval, so any conclusion drawn from C′(x)=0 is unreliable.
(d) Even though C′(5)=0, there could be interior points where C′(x) is undefined that produce a smaller cost.