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?

Question

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?

anonymous · Answer

$$(1)^2=x^2+2xy+y^2$$
$$x^2+y^2=1-2xy$$
min

anonymous · Answer

$$-2y-2x=0$$
$$x=y$$
max 1/2
min ?

stacey · Answer

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.