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So if we get zero, do we just keep using the next derivative until we get a non-zero number? What if we're in an exam and get a question that takes ages to find the derivative that gets us a non zero number?? OR, what if there is in fact no inflection point but we keep getting zero, do we...

The third derivative test is not applicaable for all inflection points. Consider f(x) = x^5 f''(0) = 0 f'''(0) = 0, and since the rule is that if f''' does not equal to zero, then it's a point of inflection. But it doesn't work in this case because (0,0) is a point of inflection. So...