Inflection question Suppose that the second derivative of a function is continuous, is positive on x <2, is negative on x > 2, and is zero at x = 2. What can you say about the behavior of the function at x = 2?
A. f is a maximum at x = 2
B. f is a minimum at x = 2
C. f changes inflection at x = 2
D. There is not enough information given.
I think the answer for this is C but im not really sure. If the second derivative changes between 2 ( before 2 and after) then its changing from concave up to concave down which means there is an inflection change. Is C the right answer?
 A: Yes, C is correct. By definition, an inflection point is one where the concavity (second derivative) changes. (In all this I assumed that you mis-copied C, which probably said "f has an inflexion point at 2.")
A: We know what you said about the second derivative. So what does this imply for the first one?
Well before $x=2$, the function must be rising! why, because its derivative is positive up to $x=2$ (the 2nd derivative is the first derivative of the first derivative). At $x=2$, it must be stationary. After $x=2$, it must be decreasing as its derivative is negative. 
So now this tells that the first derivative has a local maximum at $x=2$, but it doesn't tell us if the maximum is at a positive or a negative (y is greater or lesser than 0). It could be at any arbitrary y value! 
So for the actual function,it might not be flat at all at $x=2$ !!! Instead we know for sure that the slope after $x=2$ starts "curling" to the opposite direction from the direction it was curing from, and note, it may have never even become tangential to the x-axis (horizontal)! So if the function's line was the trace of a boat, before $x=2$ it may have been turning left and after $x=2$ it must have started turning right - that's all we can know!
So to your question: clearly the only thing we can deduce from the given information is that there is an inflection at $x=2$
Have a nice day :)
