Answer
FALSE.
Work Step by Step
Take $$z=x+y+1$$.
The first order partial derivatives are:
$$ \frac{ \partial z}{\partial x} = \frac{ \partial z}{\partial y} = 1 $$
Yet, the function is clearly not of the form:
$$ z = c(x+y) $$
Thus, we can conclude that the given statement is $ \fbox{false} $.
