A function is continuous on the open interval and differentiable on
MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
A function g is continuous on the open interval (0,2) and differentiable on (0,2). Its derivative is given by g′(x)=(x−1)(x−2). Which statement about the existence of absolute extrema for g on (0,2) is most accurate?
(a) g fails to have any absolute extrema on (0,2), because the interval is open and neither endpoint is included.
(b) Because the Extreme Value Theorem applies to any continuous function, g must attain both an absolute maximum and an absolute minimum somewhere in (0,2).
(c) Although the Extreme Value Theorem does not apply on an open interval, g attains an absolute maximum at x=1 but no absolute minimum on (0,2).
(d) Since g′(x) changes from negative to positive at x=1, the point x=1 is an absolute minimum while no absolute maximum exists.