Constrained optimisation to unconstrained using trigonometry

Question

I have this problem:

$max\{ xy: x^2+y^2 \leq 1, x,y >0\}$

Solving this gives us $(x,y) = (\frac{\sqrt2}2 , \frac{\sqrt2}2)$

I was wondering if we could make this somehow an unconstrained problem.

My first try was to take $x = a\cos(\theta)$ and $y = b\sin(\theta)$ with $0<a,b \leq 1$ and $ 0 \leq\theta \leq \pi/2$

But then I realised this is also constrained.

Could I have some hint as to how to proceed?

Thank you!

Answer 1

we have $$f(x)=x\sqrt{1-x^2}$$ then we have $$f'(x)=-{\frac {2\,{x}^{2}-1}{\sqrt {- \left( x-1 \right) \left( x+1 \right) }}} $$ solving $$f'(x)=0$$ we get $$x=\frac{\sqrt{2}}{2}$$ (note that $$x>0$$ is given!)