## Thomas' Calculus 13th Edition

$(\pm \sqrt 5, \sqrt 5)$ and $(\pm \sqrt 5,-\sqrt 5)$
Consider $f(x,y) =xy$ $g(x,y)=x^2+y^2-10=0$ ...(1) Consider the gradient equation as follows: $\nabla =\lambda \nabla g$ Here, $y=2 \lambda x$ or, $y=\pm \dfrac{1}{2}$ Now, $x=2 \lambda y$ or, $x=\pm y$ From equation (1), we have $g(x,y)=x^2+y^2-10=0$ This implies that $x=\pm \sqrt 5$ and $y= \pm \sqrt 5$ Points: $(\pm \sqrt 5, \sqrt 5)$ and $(\pm \sqrt 5,-\sqrt 5)$