Answer
$(\pm \sqrt 5, \sqrt 5) $ and $(\pm \sqrt 5,-\sqrt 5)$
Work Step by Step
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 $(\pm \sqrt 5,-\sqrt 5)$
