Solve partialL/partialL=w-2/3lambdaL^-frac{1{3}}K^frac{1{3}}

$$\frac { \partial L } { \partial L } = w - \frac { 2 } { 3 } \lambda L ^ { - \frac { 1 } { 3 } } K ^ { \frac { 1 } { 3 } }$$

Solve for K

$$\left\{\begin{matrix}K=-\frac{27\times \left(\frac{1-w}{\lambda }\right)^{3}L}{8}\text{, }&\lambda \neq 0\text{ and }L\neq 0\\K\in \mathrm{R}\text{, }&w=1\text{ and }\lambda =0\text{ and }L\neq 0\end{matrix}\right.$$

Solve for L

$$\left\{\begin{matrix}L=-\frac{8\times \left(\frac{\lambda }{1-w}\right)^{3}K}{27}\text{, }&\lambda \neq 0\text{ and }K\neq 0\text{ and }w\neq 1\\L\neq 0\text{, }&\sqrt[3]{K}\lambda =0\text{ and }w=1\end{matrix}\right.$$