Use lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (if an answer does not exist, enter dne.) f(x, y, z, t) = x + y + z + t; x2 + y2 + z2 + t2 = 49

You have Lagrangian



with partial derivatives (set equal to 0)







Solving the first four equations for , respectively, we can substitute these solutions in terms of into the fifth equation to find , which in turn will lead to . Denoting by , we have



and substituting into the fifth equation yields




From either choice of we arrive at , i.e. exactly two critical points at , for which we get a maximum value of 14 and minimum value of -14, respectively.