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(x1, x2, xn) = x1 + x2 + + xn; x12 + x22 + + xn2 = 36

The Lagrangian is

with partial derivatives (set equal to 0)

for each .

Let be the sum of all the multipliers ,

We notice that

so that

We know that , so

Solving the first equations for gives

and in particular

It follows that

which gives us

That is, we've found two critical points,

At the critical point with positive signs, attains a maximum value of

and at the other, a minimum value of