$x^4=0$ is a example of a curve where there is a point that has $f''(x)=0$ but is not a point of inflection (POI). This point is a stationary point with $f'(x)=0$.
I am looking for a example of a curve where there is a point where $f''(x)=0$ and $f'(x) < 0$ (non stationary point) and the point is not a POI.
On
The function $f(x)=x^4-x$ has a second derivative of zero at $x=0$. However only having the second derivative being zero is not sufficient to be an inflection point. In order for the point to be an inflection point the sign of the second derivative must change when passing through the point. If we take our example function, it's second derivative is:
$$f''(x) = 12x^2$$
And this function is always positive, which means that the second derivative never crosses the $x$ axis, and only touches it at $x=0$.
Take $f(x) = x^4-x$, at $x=0$. We have $f’(x)=4x^3-1$, so $f’(0)=-1\lt 0$. And $f’’(x)=12x^2$, which is nonnegative everywhere and $f’’(0)=0$. It is not a point of inflection, since the function is always concave up.
Of course, $f(x)=x^4+x$ will give you a similar example with $f’(0)\gt 0$.
Yves Daoust’s suggestion in the comments of $f(x)=-x$ is of course also an example that meets your required conditions, though I thought you might object to that since it is “flat”...