This is a homework problem.
We need to prove that when we apply the Newton-Raphson method to strictly quadratic concave function. It will converge in one step.
How to apply this method to maximization of
$$f(x) = 4\cdot x_1 + 6\cdot x_2 - 2\cdot x_1^2 - 2\cdot x_1\cdot x_2 - 2 \cdot x_2^2$$
I did not understand how to apply method to this function? what should be the interval?
You don't need an interval. You are using Newton-Raphson to find a solution of the system $$ \eqalign{\dfrac{\partial f}{\partial x_1} &= 4 - 4 x_1 - 2 x_2 = 0\cr \dfrac{\partial f}{\partial x_2} &= 6 - 2 x_1 - 4 x_2 = 0\cr}$$ This being a linear system, it doesn't matter where your initial point is: Newton-Raphson simply solves the linear system.