Stalks of Higher direct images of structure sheaf at smooth points

Let $k$ be a field of Characteristic zero, and we will consider normal separated schemes of finite type over $k$.

Let $X$ be such a scheme and $f: Yto X$ be a proper birational map where $Y$ is a regular scheme. If $x$ is a smooth (closed) point of $X$ i.e. if $mathcal O_{X,x}$ is a regular local ring, then is it true that the stalk at $x$ of the higher direct images of $f_*$ applied to $mathcal O_Y$ are trivial i.e. is it true that $left (R^i f_* mathcal O_Yright)_x=0, forall i>0$ ?

(https://en.m.wikipedia.org/wiki/Direct_image_functor).