If \( x^{y}+y^{x}=c \) find, \( \frac{d y}{d x} \)

The network latency \( L(x)=-2 x^{2}+12 x-10 \) for data packets is studied for input size ' \( x \) '. If the latency is qual 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 hange is zero. In a network flow system, the following equations represent the flow through three nodes \( \mathrm{A}, \mathrm{B} \), and \( \mathrm{C}: 2 A+ \) \( +C=100 \) (total inflow); \( A+3 B+2 C=200 \) (flow through node B ); \( 3 A+B+4 C=300 \) (outflow from ode \( C \) )

to find the principal resistance values for optimizing power flow across the network. 1. In a distributed system, the latency \( L(x, y, z)=x^{2}+2 y^{2}+3 z^{2} \) (in milliseconds) depends on the numbe 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.

24) \( \lim _{x \rightarrow 4} \frac{2-\sqrt{x}}{x^{2}-16}= \) ? \( \begin{array}{lllll}\text { A) }-\infty & \text { B) } \frac{1}{32} & \text { C) } 0 & \text { D) }-\frac{1}{32} & \text { E) Limit yoktur }\end{array} \)

PROBLEME 1 Partie A Soit g la fonction numérique définie sur \( ] 0 ;+\infty\left[\right. \) par \( g(x)=x(1+\ln x)^{2}-1 \) 1. On admettra que g est dérivable sur \( ] 0 ;+\infty[ \) a) Calculer \( g^{\prime}(x) \) pour tout \( \left.x \in\right] 0 ;+\infty[ \) b) Vérifier que pour tout \( x \in] 0 ;+\infty\left[g^{\prime}(x)=(1+\ln x)(3+\ln x)\right. \) c) Etudier le signe de \( g^{\prime}(x) \) suivant lês valeurs de \( x \) et en déduirele sens de variation de g.

Find the area under the curve defined by \( f(x) = 2x + 1 \) between the points \( x = 1 \) and \( x = 3 \).