> Yesterday, during a discussion of concavity, inflection points, >third derivative tests, etc. in (high school) A. P. Calculus, a >student asked a question which gets more interesting as I >investigate it. He wondered if a differentiable function which, >say, is concave downward for x<0, contains a line segment from >x=0 to x=1, then is concave upward for x>1 could be said to have >inflection points or even an "inflection segment."
> ..... >______________________________________________________________________ >Jeff Myers Granville High School Granville, OH >43023 email@example.com
Having checked 19 calculus texts ranging from Thomas' 2nd edition to two reform project texts, I find a consistent definition of an inflection point occurring only at the point where f''(c) = 0 or fails to exist and where f''(x) changes sign. Since f''(x) = 0 on [0, 1], no sign change takes place, hence, no inflection point.