Quadratic EquationHard
Question
If a, b and c are distinct positive numbers, then the expression (b + c - a)(c + a - b) (a + b - c) - abc is
Options
A.positive
B.negative
C.non-positive
D.non-negative
Solution
Since, as Am > GM
⇒
>((b + c - a)(c + a - b))1/2
⇒ c >((b + c - a)(c + a -b))1/2 .....(i)
Similarly b > ((a + b - c)(b + c - a))1/2 .....(ii)
and a > ((a + b - c)(c + a - b))1/2 .....(iii)
On multiplying Equs. (i),(ii) and (iii), we get
abc > (a + b - c)(b + c - a)(c + a - b)
Hence, (a + b - c)(b + c - a)(c + a - b) - abc <0
⇒
>((b + c - a)(c + a - b))1/2 ⇒ c >((b + c - a)(c + a -b))1/2 .....(i)
Similarly b > ((a + b - c)(b + c - a))1/2 .....(ii)
and a > ((a + b - c)(c + a - b))1/2 .....(iii)
On multiplying Equs. (i),(ii) and (iii), we get
abc > (a + b - c)(b + c - a)(c + a - b)
Hence, (a + b - c)(b + c - a)(c + a - b) - abc <0
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