<s>[INST] Find the maximum possible area of the rectangle ABZP Let ABZP be a rectangle. D and C lie on AP and BZ respectively such that ABCD is a square. X and Y lie on CD and ZP respectively such that CXYZ is a square. 
Given: ar(ABCD) + ar(CXYZ) = 1
Find the maximum possible area of the rectangle ABZP.
 [/INST] Let $x=AD$ (that is, its length). Let $y=CX$.
Clearly $y<x$ and $x^2+y^2=1$. We are to maximize $f(x)=x(x+y)$. But 
$$f(x)=x(x+y)=x^2+xy=x^2+x\sqrt{1-x^2}=x^2+\sqrt{x^2-x^4}$$
so
$$f'(x)=2x-\frac{x-2x^3}{\sqrt{x^2-x^4}}=2x-\frac{1-2x^2}{\sqrt{1-x^2}}$$
The value $x_0$ of $x$ that maximizes $f(x)$ satisfies $f'(x_0)=0$. Solve the equation and substitute.
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