Chapter 13 - Partial Derivatives - 13.3 Partial Derivatives - Exercises Set 13.3 - Page 937: 23

True.

In any direction, the curve seen in a plane surface must be a line, including x and y direction, which makes its slope constant. If the partial derivatives were not constants, then the second partial derivatives wouldn't be 0 everywhere, and there would be a concavity somewhere on the surface, and this clearly doesn't happen in a plane.