The Lagrangian is
[tex]L(x,y,\lambda)=5x^2+5y^2-\lambda(xy-1)[/tex]
Set the partial derivatives equal to 0:
[tex]\begin{cases}L_x=10x-\lambda y=0\\L_y=10y-\lambda x=0\\L_\lambda=xy=1\end{cases}[/tex]
Notice that [tex]xL_x-yL_y=10x^2-10y^2=0[/tex], or [tex]x^2=y^2[/tex].
Since [tex]xy=1[/tex], it follows either [tex]x=y[/tex], and so there are two critical points at (1, 1) and (-1, -1). Both of these points give the same value of [tex]f(\pm1,\pm1)=10[/tex]. And since the Hessian matrix for [tex]f(x,y)[/tex] is positive-definite, these points correspond to minima. [tex]f(x,y)[/tex] has no maximum value on [tex]xy=1[/tex].
