Answer
False
Work Step by Step
Given: $f_{x}=x+y^{2}$ and $f_{y}=x-y^{2}$
Take the second derivative of the function with respect to $y$ keeping $x$ constant.
$f_{xy}=1$
$f_{y}=x-y^{2}$
Take second derivative of the function with respect to $x$ keeping $y$ constant.
$f_{yx}=2y$
Thus, $f_{xy} \ne f_{yx}$
Thus, the second derivative of the function does not verify Clairaut's Theorem, that is, $f_{xy} = f_{yx}$
Hence, the statement is false.
