The function f(x,yz) = 5x has an absolute maximum value and absolute minimum value subject to the constraint x- + y- + z" -5z = 5. Use Lagrange multipliers to find these values. The maximum value isThe function f(x,y,z)=7x - 3y + 2z has an absolute maximum value and absolute minimum value subject to the constraint x2 + yz + z" = 62. Use Lagrange multipliers to find these values. Find the gradient of f(x,y,z) = 7x - 3y + 2z. Vf(x,y,z) =A lidless box is to be made with a volume of 4 fto. Find the dimensions of the box that requires the least amount of material. The dimensions of the box are ft. (Type an exact answer in simplified form. Use a comma to separate answers as needed.)Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form P = f(K,L) = CK L, where K represents capital, L represents labor, and C and a are positive real numbers with 0 that will be used by the method of Lagrange multipliers in two variables that will be used in conjunction with the given constraint equation, 25K + 50L = 200, to find the values of K and L that maximize the production function