What are the minimum and maximum values?

Help using Lagrange multipliers to find the maximum and minimum values?

• Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. f(x, y) = 6x + 10y; x^2 + y^2 = 34 Maximum = ________ Minimum = ________

• Answer:

  Rewrite your constraint as x^2 + y^2 - 34 = 0. Then, f(x,y,L) = 6x + 10y + L(x^2 + y^2 - 34). Then f_x = 6 + 2Lx f_y = 10 + 2Ly f_L = x^2 + y^2 - 34. Setting the three partial derivatives to zero gives us that: 6 + 2Lx = 0 => x = -3/L 10 + 2Ly = 0 => y = -5/L x^2 + y^2 -34 = 0 => 9/L^2 + 25/L^2 = 34 => 34/L^2 = 34 => L^2 = 1 => L = 1,-1 Therefore, the minimum/maximum occur at (-3,-5) and (3,5). Plugging the values in we find that f(-3,-5) = -18 - 50 = -68 and f(3,5) = 18 + 50 = 68.