Question

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

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

Answer 1

The correct option is B

negative

For the terms, (b+c-a) (c+a-b) (a+b-c)

AM > GM [as a,b,c are distinct, AM $\ne$ GM]

$\frac{(a+b-c)+(b+c-a)+(c+a-b)}{3}$ > 3 $\sqrt{(a+b-c)(b+c-a)(c+a-b)}$

$\Rightarrow$ $\frac{a+b+c}{3}$ > 3 $\sqrt{(a+b-c)(b+c-a)(c+a-b)}$

$\Rightarrow$ ${\left(\frac{a+b+c}{3}\right)}^3$ > (a+b-c) (b+c-a) (c+a-b) -----------------(1)

Again for a,b,c

AM > GM [a $\ne$ b $\ne$ c. so, AM$\ne$ GM]

$\frac{a+b+c}{3}$ > $3\sqrt{abc}$

${\left(\frac{a+b+c}{3}\right)}^3$ > abc

Subtracting (2) from (1), we get

0 > (a+b-c) (b+c-a) (c+a-b) - abc

(or) (a+b-c) (b+c-a) (c+a-b) - abc < 0

So, option (b) is true