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

positive

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negative

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non-positive

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non-negative

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Solution

The correct option is B negative
By using $A M \geq G M$

$\begin{matrix} x + y \geq 2 \sqrt{x y} y + z \geq 2 \sqrt{y z} z + x \geq 2 \sqrt{z x} s o, (x + y) (y + z) (z + x) \geq 2 \sqrt{x y} \cdot 2 \sqrt{y z} \cdot 2 \sqrt{z x} (x + y) (y + z) (z + x) \geq 8 (x y z) n o w, p u t x = b + c - a y = c + a - b z = a + b - c o r, x + y = 2 c y + z = 2 a z + x = 2 b (2 c) (2 a) (2 b) \geq 8 (b + c) (c + a - b) (a + b - c) a b c \geq (b + c - a) (c + a - b) (a + b - c) ∴ (b + c - a) (c + a - b) (a + b - c) - a b c \leq 0 ∴ n e g a t i v e \end{matrix}$

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