The Lagrangian,
[tex]L(x,y,\lambda)=x+y+\lambda(xy-64)[/tex]
has partial derivatives (set equal to 0)
[tex]L_x=1+\lambda y=0\implies\lambda=-\dfrac1y[/tex]
[tex]L_y=1+\lambda x=0\implies\lambda=-\dfrac1x[/tex]
[tex]L_\lambda=xy-64=0[/tex]
The first two equations tell us that [tex]-\dfrac1y=-\dfrac1x\implies x=y[/tex].
Substituting this into the second equation, we have
[tex]xy-64=0\iff x^2=0\implies x=\pm8\implies\begin{cases}x=8,y=8\\x=-8,y=-8\end{cases}[/tex]
We have that [tex]x,y>0[/tex], so we only have the one critical point on the surface [tex]x+y[/tex] at (8, 8), with an extreme value of 16.
