Q.11) Inequality

Q : If a, b, c are positive real numbers, then prove that :----

(a + 1)^7 * (b + 1)^7 * (c + 1)^7  >  7^7 * a^4 * b^4 * c^4

SOLUTION :

L.H.S. = (a + 1)^7 * (b + 1)^7 * (c + 1)^7 = [(a + 1)(b + 1)(c + 1) ]^7


=> [1 + a + b + c + ab + bc + ca + abc]^7
now [1 + a + b + c + ab + bc + ca + abc]^7 > [a + b + c + ab + bc + ca + abc]^7
 {taking 1 out }

Using AM ≥ GM, for positive real numbers a, b, c, ab, bc, ca and abc we get,
(a + b + c + ab + bc + ca + abc)/7 ≥ [a^4 * b^4 * c^4]^(1/7)
=> [(a + b + c + ab + bc + ca + abc)/7 ]^7 ≥ [a^4 * b^4 * c^4]
=> (a + b + c + ab + bc + ca + abc)^7 ≥ 7^7 [a^4 * b^4 * c^4] - - - - (ii)

from (i) and (ii),
[1 + a + b + c + ab + bc + ca + abc]^7 > 7^7 [a^4 * b^4 * c^4]
=> (a + 1)^7 * (b + 1)^7 * (c + 1)^7 > 7^7 [a^4 * b^4 * c^4] - - - (Proved)

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