Solve parametric equations satisfying that the equations have positive roots: $\left\{\begin{matrix} x_{1}+x_{2}+x_{3}+...+x_{m} &=9 \\ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}{x_{m}} &=1 \end{matrix}\right.$
At this problem, I know that $m$ should be $2$. However, how can I point out that $m=2$ is the only solutions while I do not know to prove the equations have no positive roots when $m>2$?
Please help me. Thank you so much!
Based on AM HM inequality -
$\frac{x_{1}+x_{2}+x_{3}+...+x_{m}}{m} \ge \frac{m}{ \frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}{x_{m}}}$
$x_{1}+x_{2}+x_{3}+...+x_{m} \ge m^2 \,$ as ${\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}+...+\frac{1}{x_{m}}} = 1$
As $x_{1}+x_{2}+x_{3}+...+x_{m} = 9$, max value of $m$ can be $3$ and as there is equality for $m = 3$, you can find that $x_1 = x_2 = x_3 = 3$ is a solution.