Given three people with the same utility function:
$$ u_A(x_1,x_2)=u_B(x_1,x_2)=u_C(x_1,x_2)=\sqrt{x_1x_2} $$
Prove that the following allocation is Pareto efficient:
$$ x_A=(2,2),\: x_B=(3,3),\: x_C=(1,1) $$
I read my prof's answer to this question which said: The allocation is Pareto efficient as it can be shown that the marginal rates of substitution are equal.
I have only seen that this is a necessary condition of Pareto efficiency not that it is sufficient. Is that the case? If not, how can I show that the allocation is Pareto efficient?