Calculate all of the second order partial derivatives of a function? 
My attempt:
$g_x$ = 3$x^2$-6x
$g_y$ = 3$y^2$-12y
$g_{xx}$ = 6x-6
$g_{yy}$ = 6y-12
$g_{xy}$ = 0
$g_{yx}$ = 0
Are these correct? Also would I be correct in saying that $g_{xy}$ = $g_{yx}$ for all functions?

You are right for all the partial derivatives. The result that the "mixed" partial derivatives are equal for all functions is true (given some very basic conditions) and is called Clairaut's Theorem.
