Help with this problem please

**tongzilla** · October 20th 2007, 07:15 AM

$\\ \mbox{Give that,} \\ x_A , x_B >1 \\ \mbox{ and } \\ Y_A,Y_B\geqslant0 \mbox{ (but strictly positive for at least one)}$
$Y_A + Y_B = 100$

$P_A=Y_A(x_A -1) - Y_B$
$P_B=Y_B(x_B -1) - Y_A$
Problem:
$x_A \mbox{ and } x_B \mbox{ is given }$
$\mbox{Find values of }Y_A, Y_B \mbox{ such that:}$
1) Both $P_A$ and $P_B$ is always greater than zero. Are there any restictions needed on $x$ for this to be true?

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EDIT (21 OCT 07):

Let me clarify what my actual question is. Sorry for not making this clear enough early on.

The question I am asking is this:
1) What restrictions (i.e. the general rule) are needed on $x_A$ and $x_B$ so that there are some range of values of $Y_A$ and $Y_B$ to make both $P_A$ and $P_B$ positive?
2) What is the restriction (i.e. the general rule) for $x_A$ and $x_B$ such that no matter what combination of $Y_A$ and $Y_B$ is chosen (but still sums to 100), $P_A$ and $P_B$ is always negative?
3) How can you prove your answers to questions 1) and 2) ?

**angel.white** · October 20th 2007, 10:46 AM

Originally Posted by tongzilla

$\\ \mbox{Give that,} \\ x_A , x_B >1 \\ \mbox{ and } \\ Y_A,Y_B\geqslant0 \mbox{ (but strictly positive for at least one)}$
$Y_A + Y_B = 100$

$P_A=Y_A(x_A -1) - Y_B$
$P_B=Y_B(x_B -1) - Y_A$
Problem:
$x_A \mbox{ and } x_B \mbox{ is given }$
$\mbox{Find values of }Y_A, Y_B \mbox{ such that:}$
1) $P_A$ and $P_B$ is always greater than zero. Are there any restictions needed on $x$ for this to be true?

I think you could do it like this:

$Y_A=Y_B=50$
Which satisfies $Y_A + Y_B = 100$

Then $x_A, x_B > 2$

Which satisfies:
$P_A=Y_A(x_A -1) - Y_B$
because Y_A = Y_B
$P_A=Y_A(x_A -1) - Y_A$
$P_A=Y_A[(x_A -1) - 1]$
$P_A=Y_A(x_A -2)$
And because we define $x_A>2$ we know that $x_A-2>0$ which means that $Y_A(x_A-2)>0$ , which means that $P_A > 0$

And the math will be the same for the other equation as well.

**tongzilla** · October 21st 2007, 01:00 AM

EDIT (21 OCT 07):

Let me clarify what my actual question is. Sorry for not making this clear enough early on.

The question I am asking is this:
1) What restrictions (i.e. the general rule) are needed on $x_A$ and $x_B$ so that there are some range of values of $Y_A$ and $Y_B$ to make both $P_A$ and $P_B$ positive?
2) What is the restriction (i.e. the general rule) for $x_A$ and $x_B$ such that no matter what combination of $Y_A$ and $Y_B$ is chosen (but still sums to 100), $P_A$ and $P_B$ is always negative?
3) How can you prove your answers to questions 1) and 2) ?

**tongzilla** · October 21st 2007, 01:02 AM

Originally Posted by angel.white

I think you could do it like this:

$Y_A=Y_B=50$
Which satisfies $Y_A + Y_B = 100$

Then $x_A, x_B > 2$

Which satisfies:
$P_A=Y_A(x_A -1) - Y_B$
because Y_A = Y_B
$P_A=Y_A(x_A -1) - Y_A$
$P_A=Y_A[(x_A -1) - 1]$
$P_A=Y_A(x_A -2)$
And because we define $x_A>2$ we know that $x_A-2>0$ which means that $Y_A(x_A-2)>0$ , which means that $P_A > 0$

And the math will be the same for the other equation as well.

So is x_A and x_B > 2 the one and only restriction needed? Does it only work for Y_A and Y_B =50 or it also works for other values?

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