From my computations, I have found
$$K_1^{*}=K/10, L_1^{*}=L/5, K_2^{*}=9K/10, L_2^{*}=4L/5, \lambda_1=\surd(5/2K), \lambda_2=\surd(5/L)$$
My Question is: How to find extreme value function $M(K, L)$.
First order conditions were
$L_{K_1}=1/(2\surd{K_1})-\lambda_1=0$
$L_{L_1}=1/\surd{L_1}-\lambda_2=0$
$L_{K_2}=3/(2\surd{K_2})-\lambda_1=0$
$L_{L_2}=2/\surd{L_2}-\lambda_2=0$
$L_{\lambda_1}=K-K_1-K_2=0$
$L_{\lambda_2}=L-L_1-L_2=0$
Do I have to use Hessian determinant approach but here are 4 variables, so how to tackle it?
To solve the following problem:
\begin{eqnarray*}\max_{L_1,L_2,K_1,K_2} & \sqrt{K_1}+2\sqrt{L_1}+3\sqrt{K_2}+4\sqrt{L_2}\\ \text{s.t.} & K_1+K_2=K, \ L_1+L_2=L \\ \text{and } & K_1\geq 0, \ K_2\geq 0, \ L_1\geq 0, \ L_2\geq0\end{eqnarray*}
we can split this problem into two problems:
\begin{eqnarray}\max_{L_1,L_2} & \ 2\sqrt{L_1}+4\sqrt{L_2}\\ \text{s.t.} & L_1+L_2=L \\ \text{and } & L_1\geq 0, \ L_2\geq0\end{eqnarray} and \begin{eqnarray}\max_{K_1,K_2} & \ \sqrt{K_1}+3\sqrt{K_2}\\ \text{s.t.} & K_1+K_2=K \\ \text{and } & K_1\geq 0, \ K_2\geq0\end{eqnarray}
Solutions to the above problems satisfy the following conditions:
\begin{eqnarray}\dfrac{\sqrt{L_2}}{2\sqrt{L_1}} & = & 1 \\ \dfrac{\sqrt{K_2}}{3\sqrt{K_1}} & = & 1 \\ L_1+L_2 & = & L \\ K_1 +K_2 & = & K\end{eqnarray}
Solving them we get, \begin{eqnarray}L_1 = \dfrac{L}{5}, \ L_2 = \dfrac{4L}{5} \\ K_1 = \dfrac{K}{10}, \ K_2 = \dfrac{9K}{10}\end{eqnarray}
Optimal value of the objective is: \begin{eqnarray} M(K,L)= \sqrt{10K}+\sqrt{20L} \end{eqnarray}