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

The Correct Option is B

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 \(x+y=2c,\space y+z=2a,\space z+x=2b\)
\(\therefore (2a)(2b)(2c)≥8(b+c-a)(c+a-b)(a+b-c)\)
⇒\((b+c-a)(c+a-b)(a+b-c)-abc≤0\space (negative)\)