Find the x-coordinate of each point of inflection of the graph of g on the open interval (-2, 5). Justify your answer
The Attempt at a Solution
The answer (I think) is x=0 and x=3. Well, the thing is that I asked my tutor how to justify it and he said that it is only x=0 and not 3. But when I asked my teacher for the answer he checked and said it x=0, 3, but he couldn't tell me how to justify it. So now I'm wondering if I'm right or my tutor is.
The reason why I thought it was x=0 and x=3 is because from -2, the area as you move towards 0 is increasing at a greater and greater rate, therefore it's concave up. Then after 0 it's still increasing, but its rate of increase is decreasing, therefore concave down. Then at 2 it starts decreasing but the magnitude of the rate of decrease is still increasing (therefor still concave down) until x=3. Then after 3 it's still decreasing, but the magnitude of the rate is decreasing so it's concave up again. Then at 4 it starts to increase with increasing rate, and so g is still concave up.
But I'm pretty sure that on an exam they want a quick explanation in mathematical terms, not a whole long paragraph like that. My tutor said that because at x=3 there's a cusp so that's not a solution but I don't get why.