There is a curve whose equation is given by $$y=9-x^2$$ Now find the area of largest rectangle that can be constructed in the curve. I proceeded like this$:$
I assumed the four points as $(x,y), (x,0), (-x,y), (-x,0)$ so the length of rectangle will become $2x$ and breadth will become $y$. So area$=2xy$. Or $2x(9-x^2)$ or $18x-2x^3$. Now for maximum area $\frac{dA}{dx}=0$ $\implies$ $$\frac{dA}{dx} 18x-2x^3 =0$$ $\implies$ $$18-6x^2=0$$ $\implies$ $$x=\sqrt{3}$$ Now I can simply find area. But I want to know can it be done without differentiation. As I have seen some sums like this which can be done by similarity or something like that. Just asking. Thank you.