Given a stationary point, I was taught to test if it was a maximum or a minimum using the concavity test, i.e.
If $f''(x)>0$: concave up (thus a local minimum)
If $f''(x)<0$: concave down (thus a local maximum)
But if we used the concavity test on $f(x)=x^4$ at the stationary point $x=0$ we find that $f$ is concave up everywhere except at $x=0$ because $f''(0)=0$, which is horribly inconvenient.
So my question is: does that mean that point isn't a minimum, or is the definition a bit broader than I was taught? Logically, it looks like a minimum, so is the concavity test alone not fool proof?
Graph of $y=x^4$