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)
“It is by a mathematical point only that we are wise, as the sailor or the fugitive slave keeps the polestar in his eye; but that is sufficient guidance for all our life. We may not arrive at our port within a calculable period, but we would preserve the true course.”
—Henry David Thoreau (1817–1862)
“Manners are of such great consequence to the novelist that any kind will do. Bad manners are better than no manners at all, and because we are losing our customary manners, we are probably overly conscious of them; this seems to be a condition that produces writers.”
—Flannery O’Connor (1925–1964)