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:
“We as a nation need to be reeducated about the necessary and sufficient conditions for making human beings human. We need to be reeducated not as parents—but as workers, neighbors, and friends; and as members of the organizations, committees, boards—and, especially, the informal networks that control our social institutions and thereby determine the conditions of life for our families and their children.”
—Urie Bronfenbrenner (b. 1917)
“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)
“Intuition, like the rays of the sun, acts only in an inflexibly straight line; it can guess right only on condition of never diverting its gaze; the freaks of chance disturb it.”
—Honoré De Balzac (1799–1850)