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 and/or condition:
“If courtesans and strumpets were to be prosecuted with as much rigour as some silly people would have it, what locks or bars would be sufficient to preserve the honour of our wives and daughters?”
—Bernard Mandeville (16701733)
“Now, at the end of three years struggle the nations condition is not what either party, or any man devised, or expected. God alone can claim it.”
—Abraham Lincoln (18091865)