Example of non trivial point where y''=0 but its not an inflection

The curve $y=x^4-4x$ has derivatives $$y'=4x^3-4$$ and $$y''=12x^2\ge 0\ .$$ Hence there is a possible inflection at $x=0$ but clearly it is not an inflection as $y''$ does not change sign across $x=0$ (since $y''\ge 0$ for all $x$).   

See graph, you can see it flatten out around $x=0$ (thats where $y''=0$)