Is a Cusp Considered an Inflection Point in Calculus?

[sickle]

is a point only considered an inflection point if a tangent (whether vertical or not) exists or just whether just that f(c) is continuous suffices.

For instance, is a cusp/corner point eligible for being inflection?

It seems that my textbooks (stewart vs thomas) have conflicting info (as always...><)

if it just another matter of taste?

 

[matiasmorant]

an inflection point is a point where a curve changes the sign of its curvature.
at maximums and minimums, functions do not change its curvature.

for example the curve y=Sin[x] changes its curvature when x=n*Pi, for n=...-2,-1,0,1,2...
the curve y=x^3 has an inflection point at x=0

since the sign of the curvature is always the same as the sign of the second derivative, an equivalent definition is: a point where the second derivative changes its sign (but second derivative is not the same as curvature)

you might have read, as another definition, that an inflection point is a point where f' is an extremum. which is equivalent to the definition above, since f'=extremum implies f''=0 and f'' will have a different sign at each side of the point. notice that it is f' that must be an extremum, not f.

 

[homology]

Inflection points are not quite the same as critical points of the first derivative. While critical points are those values where f'(x)=0 or f'(x) is undefined, inflection points are those points where f''(x)=0 provided f"(x) is defined in a neighborhood of the point.

So no, a cusp is not a change in concavity.