Wikipedia:Reference desk/Archives/Mathematics/2010 September 20

= September 20 =

Your article
Your sister project wikibooks' article http://en.wikibooks.org/wiki/Calculus/Extrema_and_Points_of_Inflection at the section ' The Extremum Test', why does it refer to if the (n+1)th derivative? What would be the difference of saying the (n+1)th is the first non-zero and the nth is odd, therefore it is an extremum, and just saying if the nth is the first non-zero and is even, then it is an extremum? —Preceding unsigned comment added by 24.92.78.167 (talk) 22:55, 20 September 2010 (UTC)
 * What we must do is continue to differentiate until we get, at the (n+1)th derivative, a non-zero result at the stationary point:
 * $$f^{\prime} \left(x \right)=0, \,f^{\prime \prime} \left(x \right)=0,\, \ldots ,f^{\left(n\right)} \left(x \right)=0,\,f^{\left(n+1\right)} \left(x \right)\ne 0$$
 * If n is odd, then the stationary point is a true extremum. If the (n+1)th derivative is positive, it is a minimum; if the (n+1)th derivative is negative, it is a maximum. If n is even, then the stationary point is a point of inflexion.

What aspect of the above are you unhappy with? -- SGBailey (talk) 15:06, 21 September 2010 (UTC)
 * It seems to me that the OP's complaint is nothing more than one of what we choose to label as n. Why it matters, exactly, is beyond me. --COVIZAPIBETEFOKY (talk) 16:58, 21 September 2010 (UTC)
 * It's because, for the common case where $$n=1$$, we were focused on the first derivative until we found the critical points at all. So the second derivative for identifying the type of extremum seems "extra" and we call it $$n+1$$.  --Tardis (talk) 12:37, 23 September 2010 (UTC)