Consider the function on the open interval

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the function

g(x)=x\sin\bigl(1/x\bigr)

on the open interval

(0,1)

. The function is not defined at

x=0

, and we do not extend its definition to that point. (a) Determine whether

g

is bounded on

(0,1)

. Support your answer with reasoning. (b) Decide whether the Extreme Value Theorem guarantees the existence of an absolute maximum or minimum of

g

(0,1)

. Provide a complete justification. (c) Evaluate

\displaystyle \lim_{x\to 0^{+}} g(x)

and explain how this limit informs your conclusion in part (b). (d) State whether

g

actually attains an absolute maximum or absolute minimum value on

(0,1)

. Justify your answer thoroughly.