Lagrange optimization with linear objective function and constraint

Question

I am would like to find:

$$\max U=4x_1+3x_2 \\ \text{s.t. } 2x_1+x_2 \leq 10\\ \text{and } x_1,x_2 \geq 0$$

Using Lagrange:

$$L =4x_1+3x_2 -\lambda_1(2x_1+x_2-10) + \lambda_2(x_1) + \lambda_3(x_2)$$

My idea is to use first-order conditions, for example:

$$\frac{\partial L}{\partial x_1} = 4-2\lambda_1+\lambda_2 = 0$$

. . .etc

Is this method feasible? Or does it make more sense to use linear programming to solve?

Answer 1

How about a simpler approach? We have $$U=4x_1+3x_2=2(2x_1+x_2)+x_2\leq 20+x_2.$$ Hence, in order to achieve the maximum value for $U$, we should make $x_2$ as large as possible while keeping in mind that $x_1\geq 0$. With $2x_1+x_2\leq 10$, this leads to $x_2=10$ and $x_1=0$, which gives $U=30$.