How is lambda calculated in this lagrangian?

Solution 1:

You have to write donwn the score equations:

$$ \begin{aligned} \frac{\partial \mathcal{L}}{\partial c_1} &= \frac{\sqrt{c_2}}{2\sqrt{c_1}}-\lambda = 0 \Leftrightarrow \sqrt{\frac{c_2}{c_1}} = 2\lambda \\ \frac{\partial \mathcal{L}}{\partial c_2} &= \frac{\sqrt{c_1}}{2\sqrt{c_2}}-\frac{\lambda}{1.05} = 0 \Leftrightarrow \sqrt{\frac{c_1}{c_2}} = \frac{2\lambda}{1.05} \\ \frac{\partial \mathcal{L}}{\partial \lambda} &= 100-c_1-\frac{c_2}{1.05} = 0 \Leftrightarrow c_1+\frac{c_2}{1.05} = 100\\ \end{aligned} $$

SO, it seems that there is a typo in the original document; it should be $2\lambda$, but the solution $(c_1, c_2)$ will be the same: $c_1 = 50, c_2 = 52.5$