1. The problem statement, all variables and given/known data f is a continuous function on [0, 8] and satisfies the following: Second Derivative Sign Test x ; f'' 0 [tex]\leq[/tex] x < 3 ; - 3 ; 0 3 < x < 5 ; + 5 ; Does Not Exist 5 < x < 6 ; - 6 ; 0 6 < x [tex]\leq[/tex] 8 ; - Based on this information is there a point of inflection at x = 5? (a) Definitely (b) Possibly (c) Definitely not 2. Relevant equations Definition: A point of inflection is where the concavity changes and there is a tangent line. Concavity changes when f'' changes signs. Negative f'' means concave down; positive f'' means concave up. 3. The attempt at a solution There is a sign change at f''(5), but the correct answer could either be "definitely" or "possibly." The point of inflection does not exist, but there is still a sign change, meaning it is a point of inflection. But my class and I were wondering whether it was possible that because there is no derivative at x = 5, there could also possibly be or not be a point of inflection there, too. The book says it's "possibly" a point of inflection, but going by the definition, we just presumed it to "definitely" be one.