5. It is desired to find the dimensions of the rectangle with the largest area that can be inscribed in the parabola of equation y = 4 -x and the x-axis as shown in the following figure: X If x and y represent the dimensions of the inscribed rectangle shown, then using the method of Lagrange multipliers, the Lagrangian L corresponds to A) L(x, y, A) = xy - X(x2 + 4y - 16). B) L(x, y, A) = 2xy - A(x2 + y - 4). C) L(x, y, A) = xy - X(x2 + 2y - 8). D) L(x, y, A) = xy - X(2x2 + 2y - 8).6. Below is a region Ray, in the XY plane: 1=2+3 Rry Consider the change of variable u = xy and v = x - y and the integral 1 = (x2 - y?) . ezzy-zy dAry Ray If Ruy is the region in the UV plane generated with the previous change of variable, then it is true that: A) I = vVv2 + 4u . endAur Ruv B) I = v . eud Awv Ruv C) 1 = -vVv2 + 4u. end Auv D) 1 = -v . euv dAux7. Consider the surface S of equation z = - An equation for the tangent line to S at the point P (-1, 1,; ), in the direction of the vector w = (-1, -1), corresponds to: 1 A) (x, y, z) = -1, 1, +t . con tER 2V2' 2V2' 2V2 B) (x,y, z ) = -1, 1, +t. ( -1, -1, con tER 4V C) (x, y, =) = (-1,1, +t'V24V2 con tER D) (x, y, z) = (-1,1, +t -1,-1, - con tER