1. If $5 x_{1}+6 x_{2}-7 x_{3}+4 x_{4}=1$, then the minimum value of $3 x_{1}^{2}+2 x_{2}^{2}+5 x_{3}^{2}+x_{4}^{2}$ is ).
A. 1
B. $\frac{1}{2}$
C. $\frac{15}{782}$
D. $\frac{11}{250}$
1. $\mathrm{C}$

By the Cauchy-Schwarz inequality, we know that $\left(5 x_{1}+6 x_{2}-7 x_{3}+4 x_{4}\right)^{2} \leqslant\left(3 x_{1}^{2}+2 x_{2}^{2}+5 x_{3}^{2}+x_{4}^{2}\right)$ $\left(\frac{5^{2}}{3}+\frac{6^{2}}{2}+\frac{7^{2}}{5}+\frac{4^{2}}{1}\right)$. Therefore, the minimum value of $3 x_{1}^{2}+2 x_{2}^{2}+5 x_{3}^{2}+x_{4}^{2}$ is $\frac{15}{782}$, hence the answer is $\mathrm{C}$.