An inflection point is a point on a curve at which the sign of the curvature (i.e. the concavity) changes.
According to Wikipedia,
"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 an undulation point. . . . An example of such an undulation point is y = x^4 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."
Q. 1) How does the second part 'follow from the definition' in the first part?
Q. 2) According to my textbook,
"A curve y = f(x) has one of its points x = c as an inflection point if f"(c) = 0 or is not defined; and f"(x) changes sign as x increases through c. The latter condition may be replaced by f'''(c) ≠ 0 when f'''(c) exists."
I didn't understand how the 'latter condition may be replaced'. Has it got anything to do with Wikipedia's 'sufficient definition' part? What if the 'lowest-order (above the second) non-zero derivative' is not the third but, say, the fifth derivative? In that case, is my textbook wrong? (Please give an example of such a function!)