Question

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

Answer 1

The Lagrangian is

$L(x,y,\lambda)=5x^2+5y^2-\lambda(xy-1)$

Set the partial derivatives equal to 0:

$\begin{cases}L_x=10x-\lambda y=0\\L_y=10y-\lambda x=0\\L_\lambda=xy=1\end{cases}$

Notice that $xL_x-yL_y=10x^2-10y^2=0$, or $x^2=y^2$.

Since $xy=1$, it follows either $x=y$, and so there are two critical points at (1, 1) and (-1, -1). Both of these points give the same value of $f(\pm1,\pm1)=10$. And since the Hessian matrix for $f(x,y)$ is positive-definite, these points correspond to minima. $f(x,y)$ has no maximum value on $xy=1$.