Help with Lagrangian multiplier method

Question

How do we apply Lagrangian multiplier method for a problem with more than one condition? Here is my work so far, for the below optimization problem:

Answer 1

The Lagrange multipliers schema states that if $\Omega\subset\mathbb{R}^n$ is open, and $f:\Omega\to\mathbb{R}$ is differentiable, and if $m\leq n$ and $g_1,g_2,\ldots,g_m:\Omega\to\mathbb{R}$ are continuously differentiable, then at every local extremum $a\in U$ of $f$ subject to the constraint $g_i=0$ for $i=1,2,\ldots,m$ there exist $\lambda_1,\lambda_2,\ldots,\lambda_m$ such that $\nabla f(a) = \sum_{i=1}^m\lambda_i\nabla g_i(a)$, or $\{\nabla g_i(a): i=1,2,\ldots,m\}$ are linearly dependent.

Therefore, to find local extrema of $f$ subject to the constraints $g_i=0$, we need to treat separately the points where $\{\nabla g_i(a): i=1,2,\ldots,m\}$ are linearly dependent, and solve the system of $n+m$ equations in $x_1,\ldots,x_n,\lambda_1,\ldots,\lambda_m$ given by $$\frac{\partial f}{\partial x_j}=\sum_{i=1}^m\lambda_i\frac{\partial g_i}{\partial x_j}, j=1,\ldots,n;\quad g_i = 0, i=1,\ldots,m.$$