Mathematics

Problem

A student analyzing the function

g

finds that

g'(2)=0

and

g''(2)=0

. The student concludes that

x=2

is neither a relative maximum nor a relative minimum of

g

. Which of the following best evaluates the student’s reasoning? a)

\text{The conclusion is justified because }g''(2)=0\text{ implies the point is an inflection point, so it cannot be an extremum.}

\text{The conclusion is not justified because }g'(2)=0\text{ already ensures the point is an extremum, even though the second derivative test fails.}

\text{The conclusion is not justified because }g''(2)=0\text{ makes the second-derivative test inconclusive, so additional analysis is required.}

\text{The conclusion is not justified because }g''(2)=0\text{ guarantees that }x=2\text{ is a relative minimum.}