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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

If a, b, c are positive real numbers. Then prove that (a + 1)7 (b + 1)7 (c + 1)7 > 77 a4 b4 c4

Grade:11

1 Answers

Aditi Chauhan
askIITians Faculty 396 Points
7 years ago
Sol. Given that a, b, c are positive real numbers. To prove that
(a + 1)7 (b + 1)7 (c + 1)7 > 77 a4 b4 c4
Consider L. H. S. = (1 + 7)7. (1 + b)7. (1 + c)7
= [(1 + a) (1 + b) (1 + c)]7
[1 + a + b + c + ab + bc + ca + abc]7 > [a + b + c + ab + bc + ca + abc]7 . . . . . . . . . . . . . . .(1)
Now we know that AM ≥ GM using it for + ve no’s a, b, c, ab, bc, ca and abc, we get
⇒ (a + b + c + ab + bc + ca + abc)7 ≥ 77 (a4 b4 c4 ) a
From (1) and (2), we get
[(1 + a) (1 + b) (1 + c)]7 > 77 a4 b4 c4
Hence proved.

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