Explanation

(c)
Given, $f(x,y) = 2(x - y)^{2} - x^{4} - y^{4}$
On differentiating partially w.r.t. $x,\$twicely
$f_{x} = 4(x - y) - 4x^{3}$
$\Rightarrow f_{xx} = 4 - 12x^{2}$
$\Rightarrow {(f_{xx})}_{(0,0)} = 4 - 0 = 4$
Similarly, $f_{yy} = 4 - 12y^{2}$
$\Rightarrow {(f_{yy})}_{(0,0)} = 4 - 0 = 4$
and$\ f_{xy} = \ - 4 + 0$
$\Rightarrow {(f_{xy})}_{(0,0)} = \ - 4$
$\therefore\left| f_{xx}\ f_{yy}\ –f_{xy}^{2} \right|_{(0,0)} = \left| 4(4) - ( - 4)^{2} \right| = 0$