# Help with this problem please

• Oct 20th 2007, 07:15 AM
tongzilla
Help with this problem please
\$\displaystyle \\ \mbox{Give that,} \\ x_A , x_B >1 \\ \mbox{ and } \\ Y_A,Y_B\geqslant0 \mbox{ (but strictly positive for at least one)}\$
\$\displaystyle Y_A + Y_B = 100\$

\$\displaystyle P_A=Y_A(x_A -1) - Y_B\$
\$\displaystyle P_B=Y_B(x_B -1) - Y_A\$
Problem:
\$\displaystyle x_A \mbox{ and } x_B \mbox{ is given } \$
\$\displaystyle \mbox{Find values of }Y_A, Y_B \mbox{ such that:}\$
1) Both \$\displaystyle P_A\$ and \$\displaystyle P_B\$ is always greater than zero. Are there any restictions needed on \$\displaystyle 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 \$\displaystyle x_A \$ and \$\displaystyle x_B\$ so that there are some range of values of \$\displaystyle Y_A\$ and \$\displaystyle Y_B\$ to make both \$\displaystyle P_A\$ and \$\displaystyle P_B\$ positive?
2) What is the restriction (i.e. the general rule) for \$\displaystyle x_A\$ and \$\displaystyle x_B\$ such that no matter what combination of \$\displaystyle Y_A\$ and \$\displaystyle Y_B\$ is chosen (but still sums to 100), \$\displaystyle P_A\$ and \$\displaystyle P_B\$ is always negative?
3) How can you prove your answers to questions 1) and 2) ?
• Oct 20th 2007, 10:46 AM
angel.white
Quote:

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

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

I think you could do it like this:

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

Then \$\displaystyle x_A, x_B > 2\$

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

And the math will be the same for the other equation as well.
• Oct 21st 2007, 01:00 AM
tongzilla
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 \$\displaystyle x_A \$ and \$\displaystyle x_B\$ so that there are some range of values of \$\displaystyle Y_A\$ and \$\displaystyle Y_B\$ to make both \$\displaystyle P_A\$ and \$\displaystyle P_B\$ positive?
2) What is the restriction (i.e. the general rule) for \$\displaystyle x_A\$ and \$\displaystyle x_B\$ such that no matter what combination of \$\displaystyle Y_A\$ and \$\displaystyle Y_B\$ is chosen (but still sums to 100), \$\displaystyle P_A\$ and \$\displaystyle P_B\$ is always negative?
3) How can you prove your answers to questions 1) and 2) ?
• Oct 21st 2007, 01:02 AM
tongzilla
Quote:

Originally Posted by angel.white
I think you could do it like this:

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

Then \$\displaystyle x_A, x_B > 2\$

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