Answer
$(\pm \sqrt 5, \sqrt 5) $ and $(\pm \sqrt 5,-\sqrt 5)$
Work Step by Step
As we are given that $f(x,y) =xy$
and the equation of constraint $g(x,y)=x^2+y^2-10=0$
since, we have the gradient equation $\nabla =\lambda \nabla g$
Now, $y=2 \lambda x \implies y=\pm \dfrac{1}{2}$
and $x=2 \lambda y \implies x=\pm y$
Now,
$g(x,y)=x^2+y^2-10=0 \implies y= \pm \sqrt 5; x=\pm \sqrt 5$
Hence, our points are: $(\pm \sqrt 5, \sqrt 5) $ and $(\pm \sqrt 5,-\sqrt 5)$
