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) = 5x2 + 5y2; xy = 1

The Lagrangian is



Set the partial derivatives equal to 0:



Notice that , or .

Since , it follows either , and so there are two critical points at (1, 1) and (-1, -1). Both of these points give the same value of . And since the Hessian matrix for is positive-definite, these points correspond to minima. has no maximum value on .