JEE Main & Advanced Mathematics Equations and Inequalities Question Bank Self Evaluation Test - Linear Inequalities

If, a, b, c, distinct positive real numbers then the expression \[\left( b+c-a \right)\text{ }\left( c+a-b \right)\text{ }\left( a+b-c \right)-abc\] is

A) Positive

B) Negative

C) Non-positive

D) Non-negative

Correct Answer: B

Solution :

[b] \[(b+c-a)(c+a-b)(a+b-c)-abc\]

Without loss of generality we can assume,

\[a>b>c\]

Applying A.M, G.M. pair wise

\[(b+c)>2\sqrt{bc}\] (i)

\[(a+c)>2\sqrt{ac}\] (ii)

\[(a+b)>2\sqrt{(ac)}\] (iii)

Multiplying equation (i), (ii), (iii), we get \[(a+b)(b+c)(c+a)>8abc\]

Let us put \[b+c=2p;c+a=2q;a+b=2r\]

\[\Rightarrow a=-p+r+q;b=p-q+r;c=p+q-r\]

\[\Rightarrow 2p.2q.2r\ge r\ge 8(q+r+p);(p+r-q);(p+q-r)\]

\[\Rightarrow pqr\ge (q+r-p)(p+r-q)(p+r-r)\]

Replacing p, q, r by a b c

Without loss of generality, we get

\[abc\ge (a+b-c)(b+c-a)(c+a-b)\]

\[\Rightarrow \] The required expression is always negative