Correct Answer :- A
Explanation : f(x,y,z) = g(x^{2} + y^{2} - 2z^{2}).
df'/dx = g'(x^{2} + y^{2} - 2z^{2}) (2x)
df"/dx” = g"(x^{2} + y^{2} - 2z^{2}) (4x^{2}) + g'(x^{2} + y^{2} - 2z^{2})*2.........(1)
df/dy = g'(x^{2} + y^{2} - 2z^{2}) (2y)
df"/dy” = g"(x^{2} + y^{2} - 2z^{2}) (4y^{2}) + g'(x^{2} + y^{2} - 2z^{2})*2.........(2)
df'/dz = g'(x^{2} + y^{2} - 2z^{2}) (2y)
df"/dz” = g"(x^{2} + y^{2} - 2z^{2}) (4z^{2}) + g'(x^{2} + y^{2} - 2z^{2})*2.........(3)
Adding (1), (2) and (3)
g"(x^{2} + y^{2} - 2z^{2})(4x^{2} + 4y^{2} + 16z^{2}) + g'(x^{2}+ y^{2} - 2z^{2}) (2 + 2 - 4)
= 4(x^{2} + y^{2} + 4z^{2}) g"(x^{2} + y^{2} - 2z^{2})