60. \( f(x)=\left\{\begin{array}{c}2 x^{2}-1, \text { if } x \leq 1 \\ 4 x-1, \text { if } x>1\end{array}\right. \), then the derivative of \( f \) at \( x=4 \) is A. 4 B. 3 C 0 D. does not exist 61. What is the maximum possible area of a rectangle in square units with diagonal of length 16 units ? A. 48 B. 64 C. 128 D. 256 62. The maximum profit that a company can make if the profit function company is given by \( P(x)=36+ \) \( 72 x-18 x^{2} \) is: A. 2,232 B. 108 C. 124 D. 31

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.

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.

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

Use Lagrange interpolation polynomial of degree 3 to estimate \( f(8.4) \) if \( f(8.1)=16.9441, f(8.3)=17.5649, f(8.6)=18.5052 \), and \( f(8.7)=18.8209 \) Verify that \( \sum_{i=0}^{3} L_{i}(8.4)=1 \)

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.