Use Lagrange multipliers to find the given extremum. Assume that x and y are positive. Minimize f(x, y) = x2 − 4x + y2 − 16y + 42 Constraint: x + y = 24   Minimum of f(x, y) =   at (x, y) =

Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive. Minimize f(x, y) = x2 + y2 Constraint: x + 2y − 5 = 0   f(______) = ________

Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.                                                                                                       Minimize f(x, y) = x2 − y2 Constraint: x − 2y + 6 = 0

Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.                                                                                                  Minimize f(x, y) = 2x + y Constraint: xy = 32

Use Lagrange multipliers to find the indicated extrema, assuming that x and y are positive.                                                                                                     Minimize f(x, y) = x2 + y2 Constraint: x + y − 8 = 0