Question

If $x,y\in R,$ such that $x+2y=1$, find the maximum value of $xy.$

Answer 1

$x+2y=1,x,y\in R$

$x=(1-2y)$

$A=xy$

$A=(1-2y)y=y-2y^2$

$\frac{dA}{dy}=1-4y\Rightarrow formaxvalue1-4y=0\Rightarrow y=\frac{1}{4}\therefore x=1-2\left(\frac{1}{4}\right)=1-\frac{1}{2}=\frac{1}{2}$

So, the max. value of $xy=\left(\frac{1}{2}\right)\left(\frac{1}{4}\right)=\frac{1}{8}$