A Necessary But Not Sufficient Condition
If x is an inflection point for f then the second derivative, f″(x), is equal to zero if it exists, but this condition does not provide a sufficient definition of a point of inflection. One also needs the lowest-order (above the second) non-zero derivative to be of odd order (third, fifth, etc.). If the lowest-order non-zero derivative is of even order, the point is not a point of inflection, but a undulation point. However, in algebraic geometry, both inflection points and undulation points are usually called inflection points. An example of such a undulation point is y = x4 for x=0.
It follows from the definition that the sign of f′(x) on either side of the point (x,y) must be the same. If this is positive, the point is a rising point of inflection; if it is negative, the point is a falling point of inflection.
Read more about this topic: Inflection Point
Famous quotes containing the words sufficient condition, sufficient and/or condition:
“History suggests that capitalism is a necessary condition for political freedom. Clearly it is not a sufficient condition.”
—Milton Friedman (b. 1912)
“In making the great experiment of governing people by consent rather than by coercion, it is not sufficient that the party in power should have a majority. It is just as necessary that the party in power should never outrage the minority.”
—Walter Lippmann (1889–1974)
“... solitude is such a potential thing. We hear voices in solitude, we never hear in the hurry and turmoil of life; we receive counsels and comforts, we get under no other condition ...”
—Amelia E. Barr (1831–1919)