i really dontknow this questoin any help would be great
prove that if real numbers a,b,c satisfy
a + b + c>0, ab +ac + bc>0 , abc>0
then each of a,b,c is positive
thank you in advance
It ain't elegant but it works:
$\displaystyle ab + ac + bc > 0$
$\displaystyle a(b + c) + bc > 0$
1) $\displaystyle b + c > -\frac{bc}{a}$ if $\displaystyle a > 0$
or
2) $\displaystyle b + c < -\frac{bc}{a}$ if $\displaystyle a < 0$
or
3) $\displaystyle bc > 0$ if $\displaystyle a = 0$
Now:
3) is the easiest case. $\displaystyle abc > 0$ and $\displaystyle a = 0$ are contradictory. Thus $\displaystyle a \neq 0$.
So consider case 1):
1) $\displaystyle b + c > -\frac{bc}{a}$ if $\displaystyle a > 0$
We know that $\displaystyle abc > 0$ and a is positive, so $\displaystyle bc > 0$. This means that either both b, c are positive or both b, c are negative.
If both b, c are positive then $\displaystyle b + c > -\frac{bc}{a}$ is trivial since the RHS is negative, and we have no contradiction.
If both b, c are negative then
$\displaystyle b + c > -\frac{bc}{a}$
$\displaystyle \frac{bc}{a} + c > -b$
$\displaystyle c \cdot \left ( \frac{b}{a} + 1 \right ) > -b$
$\displaystyle c \cdot \left ( \frac{a + b}{a} \right ) > -b$
Now if $\displaystyle a + b > 0$ then
$\displaystyle c > \left ( \frac{a}{a + b} \right ) \cdot -b$
is a contradiction because $\displaystyle c < 0$ and the RHS is negative.
If $\displaystyle a + b < 0$ and we know that c is negative then $\displaystyle a + b + c > 0$ cannot be true because $\displaystyle (a + b) + c < 0$, a contradiction.
Thus we are left with $\displaystyle a + b = 0$.
Thus
$\displaystyle c \cdot \left ( \frac{a + b}{a} \right ) > -b$
$\displaystyle c > 0$
another contradiction.
Thus both b and c cannot be negative.
Thus case 1) says that if a is positive, so are both b and c.
Case 2) works in a similar fashion to show that, at best, all three of a, b, and c must be negative. (I leave this for you to show.) This again contradicts $\displaystyle a + b + c > 0$.
-Dan
The basic idea behind what I was doing is this:
Given a real number a, we know that only one of the following possibilities is true:
a < 0
a = 0
a > 0
I used this idea in two places in the proof of case 1): First in showing that if a is positive then b,c must both be either positive or negative. I showed that there is no contradiction if both b, c are positive. Then I showed that if b,c are both negative that a contradiction exists by looking at a + b in reference to >,=,<. There was a contradiction in all three cases, so I showed that a + b, a real number, is neither >, =, or < 0, an impossibility.
-Dan