You titled this "Lagrange Method Problem". Do you know what the "Lagrange Method" is?
The "Lagrange Method" (also called the "Lagrange Multipler Method") for minimizing (or maximizing) the function \(\displaystyle F(x_1, x_2, x_3)\) subject to constraints \(\displaystyle g_1(x_1, x_2, x_3)= A\), \(\displaystyle g_2(x_1, x_2, x_3)= B\), etc. is to minimize (or maximize) \(\displaystyle F(x_1, x_2)- \lambda_1 g_1(x_1, x_2, x_3)- \lambda_2 g_2(x_1, x_2, x_3)+ \cdot\cdot\cdot\). Take the derivatives with respect to \(\displaystyle x_1\), \(\displaystyle x_2\), \(\displaystyle x_3\) and all of the "\(\displaystyle \lambda\)"s and set them equal to 0. With n constraints, so n "\(\displaystyle \lambda\)"s, that gives n+ 3 equations to solve for \(\displaystyle x_1\),\(\displaystyle x_2\), \(\displaystyle x_3\), and the "\(\displaystyle \lambda\)"s. Since specific values for the "\(\displaystyle \lambda\)"s is not necessary for a solution, it is often simplest to eliminate them by dividing one equation by another.
