Does a point of inflection occur if the second derivative graph does not have a derivative at the point, but the first derivative has a max/min at the point?
In terms of f'(x), a point of inflection occurs where f'(x) has a max/min, like you said. To solve for this you take the derivative of f'(x), which is f''(x) and set it equal to zero. So if you are looking at the f''(x) graph then you need to be looking for when f''(a)=0.
Ok, thank you. Makes more sense now. So you are correct that the second derivative doesn't exist at x=0, but this isn't a requirement for an inflection point. If a is an inflection point then f'(a) exists and f''(a) changes sign at x=a. So if you show that the sign of f''(x) changes at x=0 and f'(0) exists you have an inflection point. You can use the graph to justify that before x=0, the f'(x) graph has a negative slope while after x=0 it has a positive one. Sometimes the AP tests get picky about topics and want you to answer a certain way so I'd check with your teacher or look up the answer when you can to make sure that they agree with this reasoning.