1. Let $\lambda$ be a given positive number. Find the largest constant $c(\lambda)$ such that for any $x_{1}, x_{2} \geqslant 0$, we have $x_{1}^{2}+x_{2}^{2}+\lambda x_{1} x_{2} \geqslant c\left(x_{1}+x_{2}\right)^{2}$.
(Tip: In the original inequality, let $x_{1}=x_{2}=1$. Get $c \leqslant$ $\frac{\lambda+2}{4}$. Then consider whether $c=\frac{\lambda+2}{4}$ holds for the original inequality. Consider two cases: $1^{\circ}$ When $\lambda<2$. Use the comparison method to prove $x_{1}^{2}+x_{2}^{2}+\lambda x_{1} x_{2}$ $\geqslant \frac{\lambda+2}{4}\left(x_{1}+x_{2}\right)^{2}$. Thus, we get $c_{\text {max }}=\frac{\lambda+2}{4} ; 2^{\circ}$ When $\lambda \geqslant 2$, from $x_{1}^{2}+x_{2}^{2}+\lambda x_{1} x_{2} \geqslant x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}=\left(x_{1}+x_{2}\right)^{2}$ we get $c_{\text {max }}=1$.)