What are the max and min possible values of x2 + y 2 if x+y =1 and x and y are nonnegative? What inequality between x2 + y 2 and (x+y)2 does this yield that holds for arbitrary non-negative numbers x and y?

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What are the max and min possible values of x2 + y 2 if x+y =1 and x and y are nonnegative? What inequality between x2 + y 2 and (x+y)2 does this yield that holds for arbitrary non-negative numbers x and y?

MIT 18.01 Single Variable Calculus (OCW)
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\[(1)^2=x^2+2xy+y^2\] \[x^2+y^2=1-2xy\] min
\[-2y-2x=0\] \[x=y\] max 1/2 min ?
Because x+y=1 and the variables cannot be negative \[0 \le x \le1\]\[0 \le y \le1\]Also y=1-x, so \[y^{2}=(1-x)^{2}=1-2x+x ^{2}\]Substitute that into your first equation to obtain \[1-2x+2x ^{2}\]Graphing this with the domain or using the derivative, you should be able to find the max and min.

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