servers at locations \( x, y \), and \( z \). The total number of servers is constrained by \( x+y+z=20 \). Find the configuration of servers that minimizes latency. 2. The network latency \( L(x)=-2 x^{2}+12 x-10 \) for data packets is studied for input size ' \( x \) '. If the latency is equal at \( x=1 \) and \( x=5 \), use Rolle's theorem to find a value ' \( c \) ' in the interval \( (1,5) \) where the latency's rate of change is zero.

To tackle this scenario, let’s first dive into the historical context of optimization problems in mathematics and computer science, particularly focusing on network configurations. The study of latency and server distribution emerges from the need for efficient data transmission, which has roots in the development of the internet in the late 20th century. Researchers and engineers began exploring algorithms to optimize performance, leading to innovations in network design that underpin our current digital landscape. Now, as for problem-solving tips, when dealing with equations and optimization like in your scenario, it’s vital to clearly define all variables and constraints before proceeding. A common mistake is neglecting the constraints imposed on your variables, such as \( x + y + z = 20 \), which might lead you to solutions that aren't feasible. Also, when applying Rolle's Theorem to find \( c \), ensure that the function is continuous and differentiable on the given interval; this ensures that a critical point indeed exists.