if x+y=1 then what is the min. value of x^3+y^3+xy ?

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- Feb 23rd 2013, 04:26 AMRuyHayabusaMinimium value
if x+y=1 then what is the min. value of x^3+y^3+xy ?

- Feb 23rd 2013, 05:14 AMMINOANMANRe: Minimium value
Min =1/2 when x =1/2

- Feb 24th 2013, 02:47 AMRuyHayabusaRe: Minimium value
Step-by-step solution please.

- Feb 24th 2013, 03:35 AMProve ItRe: Minimium value
- Feb 24th 2013, 06:27 AMHallsofIvyRe: Minimium value
Another way to do it: , .

We must have , for some number , as well as x+y= 1.

So which says that .

If x= y, from x+y= 1 we have x= y= 1/2.

If then so we can divide by it to get 3(x+ y)- 1= 0 which says x+y= 1/3 contradicting x+y= 1.

Thus x= y= 1/2 is the only solution. - Feb 24th 2013, 07:04 AMHallsofIvyRe: Minimium value
Yet

**another**way, and probably simplest:

From x+ y= 1, y= 1- x. Putting that into [tex]x^3+ y^3+ xy[/itex] as Prove It suggested, we have . We can write that as and, completing the square, . Since a square is never negative, that will have its minimum value when x= 1/2.