Question

If $a, b$ and $c$ are distinct positive numbers, then the expression $(b + c - a ) (c+ a - 6) (a + b - c) - abc$ is

• A. positive
• B. negative
• C. non-positive
• D. non-negative

Answer 1

Answer: (B)

negative

Answer 2

The correct answer is B:Negative
Given that;
$a,b,c$ are distinct positive numbers
By using $AM≥GM$
$x+y≥2\sqrt{xy}$
$y+z≥2\sqrt{yz}$
$z+x≥2\sqrt{zx}$
So, $(x+y)(y+z)(z+x)≥2\sqrt{xy}.2\sqrt{yz}.2\sqrt{zx}$
⇒$(x+y)((y+z)(z+x)≥8(xyz)$
Now let $x=b+c-a,\space y=c+a-b,\space z=a+b-c$
or $x+y=2c,\space y+z=2a,\space z+x=2b$
$\therefore (2a)(2b)(2c)≥8(b+c-a)(c+a-b)(a+b-c)$
⇒$(b+c-a)(c+a-b)(a+b-c)-abc≤0\space (negative)$