MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation
Problem
The function f(x)=xlnx−x is defined for x>0. When seeking the absolute minimum of f on (0,∞), which of the following best explains why evaluating endpoints is unnecessary?
(a) Because the Extreme Value Theorem does not apply to open intervals, absolute extrema cannot occur at endpoints, so they can be ignored.
(b) Because x→0+limf(x)=−∞, the function has no lower bound, so evaluating endpoints is pointless.
(c) Because the interval has no finite endpoints, any absolute extrema must occur at interior critical points.
(d) Because lnx is increasing on (0,∞), f must also be increasing on that interval.