Answer
True.
Work Step by Step
From the local approximation equation, we know that
\[
f_{x}\left(x_{0}, y_{0}\right) d x+f_{x}\left(x_{0}, y_{0}\right) d y=d z
\]
Therefore, that means it represents a plane:
\[
0=f_{x}\left(x_{0}, y_{0}\right) \hat{i}+f_{x}\left(x_{0}, y_{0}\right) \hat{j}-\hat{k}
\]
Thus, it's the equation of a normal vector which passes through $\left(x_{0}, y_{0}, f\left(x_{0}, y_{0}\right)\right)$
