Yes, that is what it is referring to. It is similar to verifying maximums and minimums using the second derivative test.
So the question indicates: Write a statement/proof that ifwe have a point c with f′′(c) = 0, we can sometimes use the third derivative to determinewhether f has an inflection point at c, and of which type.
So, I understand how to write a statement and proof of how one can use the third derivative to determine if f has an inflection point (although only in certain cases). Now I'm not sure what the question 'and of which type' refers to. Is it referring to whether the curve goes concave up to concave down or goes concave down to concave up?
So basically, if c is an inflection point, then if f'''(c)<0, the curve goes from concave up to concave down, and if f'''(x)>0 then the curve goes from concave down to concave up (just to get the basic idea, not meant to be completely formal)
Just had another quick question. Do you think it would be better to have two separate theorems, with one indicating whether or not the point is an inflection point, and the second determining its concavity? If I put it all into one theorem, then I end up repeating the statement 'then (c, f(c)) is an inflection point' both when f'''(x)>0 and f'''(x)<0.