4.1 [10 E-iRI+Re subject to the constraint i +i-I.given currents i and ia by the method of Lagrange multipliers points) An electrical current / flows through a circuit containing the resistors Rs and in parallel. The currents i and i through the individual resistors minimize energy loss by Kirchhoff's circuit law (The algebraic sum of currents in a network of conductors meeting at a point is zero.). Find the 4.2 [10 points) The company sells two products that are partial substitutes for each other, such and tea. If the price of one product rises, then the demand for the other product rises. The quantities demanded, g and ga. are given as a function of the prices, Ph and pa, by as coffee 91-517 _ 3.5 Pi + 0.8Pa, -770-4.4Pa + 1.4 Pi. company charge in order to maximize the total sales revenue ? 4.3 [10 points Design a closed cylindrical container which holds 200 cm3 and has the minimal possible surface area. What should its dimensions be? 4.4 [10 points] The function f(x,v) = sinx cos y has a critical point at Go) what sort of critical point is it ? 4.5 [10 points] Find the critical points of the function f(z,y)-ã§3-3x+ 3-3y and dassify them as local maxima, local minima, saddle points, or none of these.