For a polynomial $f$, if we know that $f'(a)=0$, and that $f''(a)=0$, would the point $(a, f(a))$ be considered a point of inflection? I do not think it would, but I am having trouble visualizing it.
Edit: What I am really asking is, given $f'(a)=0$, if we know that $f''(a)>0$ implies a minimum, and that $f''(a)<0$ implies a maximum, does $f''(a)=0$ directly imply anything? Is there any particular significance to that point of the graph like there is to minimums, maximums, points of inflection. etc?
Not necessarily. A point of inflection is where $f''(a)$ changes sign.
Simply knowing that $f''(a)=0$ does not imply a point of inflection, but it is necessary to have a point of inflection.
As @mfl commented, $\displaystyle \frac{d^2}{dx^2}x^4=0$, but $f(x)=x^4$ doesn't have a point of inflection at $x=0$.
Whereas, $\displaystyle \frac{d^2}{dx^2}x^5=0$ and it does have a point of inflection at $x=0$.