- #1
anemone
Gold Member
MHB
POTW Director
- 3,883
- 115
Hi MHB,
I've one problem that I think I've already solved half of it, but fact is I really don't know if I am on the right track... that problem is hurting my head so much...
Problem:
For all positive real $x$ and $y$, find the minimum of $x+y+xy$ and $x+y-xy$ if $(x+y+xy)(x+y-xy)=xy$.
Attempt:
AM-GM tells us $\dfrac{(x+y+xy)+(x+y-xy)}{2}\ge \sqrt{(x+y+xy)(x+y-xy)}$ and the LHS simplifies to $x+y$ whereas the RHS simplifies to $\sqrt{xy}$ since $(x+y+xy)(x+y-xy)=xy$. Therefore, we have that $x+y\ge \sqrt{xy}$.
I then naturally think of to subtract $xy$ from both sides of the inequality above and I get
$x+y-xy\ge -xy+\sqrt{xy}$
$x+y-xy\ge -(\sqrt{xy}-0.5)^2+0.25$
Hence, the minimum value for $x+y-xy$ is 0.25.
That's all that I managed to find, and I don't know if this is true, because I don't have the answer to the problem, and I don't know how to proceed to find the minimum value for the other expression and I also don't know how to verify if my first conclusion is correct...
Please, if someone could help me, that would be great and many thanks in advance!![Eek! :eek: :eek:](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
I've one problem that I think I've already solved half of it, but fact is I really don't know if I am on the right track... that problem is hurting my head so much...
Problem:
For all positive real $x$ and $y$, find the minimum of $x+y+xy$ and $x+y-xy$ if $(x+y+xy)(x+y-xy)=xy$.
Attempt:
AM-GM tells us $\dfrac{(x+y+xy)+(x+y-xy)}{2}\ge \sqrt{(x+y+xy)(x+y-xy)}$ and the LHS simplifies to $x+y$ whereas the RHS simplifies to $\sqrt{xy}$ since $(x+y+xy)(x+y-xy)=xy$. Therefore, we have that $x+y\ge \sqrt{xy}$.
I then naturally think of to subtract $xy$ from both sides of the inequality above and I get
$x+y-xy\ge -xy+\sqrt{xy}$
$x+y-xy\ge -(\sqrt{xy}-0.5)^2+0.25$
Hence, the minimum value for $x+y-xy$ is 0.25.
That's all that I managed to find, and I don't know if this is true, because I don't have the answer to the problem, and I don't know how to proceed to find the minimum value for the other expression and I also don't know how to verify if my first conclusion is correct...
Please, if someone could help me, that would be great and many thanks in advance!