$f(x) = 1+x^3$; $f'(x) = 3x^2$; $f'(0) = 0$. $f''(x) = 6x$; $f''(0) = 0$ — the second derivative test is inconclusive. $f'''(0) = 6 \neq 0$, so $x=0$ is a point of inflection. The function changes concavity at $x=0$ without a local extremum.