73 On donne la fonction \( f: \mathbb{R} \rightarrow \mathbb{R} \) \[ x \mapsto \frac{1}{2 \sin 2 x+1} \] 1. Quel est l'ensemble de définition de \( f \) ? 2. a) La fonction \( f \) est-elle paire ? impaire ? Justifier b) Démontrer que \( f \) est périodique de période \( \pi \). 3. Trouver les antécédents de \( \frac{1}{2} \) par \( f \). 4. Trouver l'image réciproque de \( \mathbb{R}^{+} \)par \( f \).

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 \( C \) )

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.

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.

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.

If time \( t \) be regarded as a function of velocity \( v \), prove that the rate decrease of acceleration \( f \) is given by \( f^{\frac{d}{} f^{\prime}} d v^{2} \) [Bikaner सस्न रेखा में चल रहे किसी कण की मूल बिंद ती दूरी \( x \) पर वेग \( \left.\left(a^{2}-x^{2}\right)^{\frac{1}{2}} \right\rvert\, x^{2} \) सम्यनुपाती है। इसके त्वरण का नियम ज्ञात कीजिए। The velocity of a particle moving along a straight line, when a distance \( x \) from the origin varies as \( \left(a^{2}-x^{2}\right)^{\frac{1}{2}} / x^{2} \). Find the law