My teacher told me (and Wikipedia backs this up) that the second derivative of a stationary point doesn't tell us anything about whether the point is a maximum, minimum or an inflection point. But I don't understand how it can be anything other than an inflection point.
Please can you:
a) explain why it can be a maximum/minimum with a second derivative of $0$
b) give an example where this happens (where there is a maximum/minimum and it has a second derivative of $0$)
Thank you.
Best Answer
Consider polynomials $x^4$ for minimum and $-x^4$ for maximum.
The intuition is that "interesting things" might happen beyond the second derivative. To give you a more curious example: $$f(x) = \begin{cases}e^{-1/x^2} & \text{ for }x\neq 0\\0&\text{ for } x=0\end{cases}$$
is a smooth function which has all its derivatives at zero equal to zero, and yet it has a minimum at that point.
I hope this helps $\ddot\smile$