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Question: If a = x + y + z + w and b = (xy + yz + zw + wx + wy + xz), then which of the statement (s), is/are ...

If a = x + y + z + w and b = (xy + yz + zw + wx + wy + xz), then which of the statement (s), is/are true

A

8a2³ 3b

B

3a2³ 8b

C

a2b³ 27

D

None of these

Answer

3a2³ 8b

Explanation

Solution

Since, 3(x2 + y2 + z2 + w2 ) – 2 (xy + yz + zx + zw + wx + wy)

= (x – y)2 + (x – z)2 + (x – w)2 + (y – z)2 + (y – w)2

+ (z – w)2 ³ 0

Ž 3∑x2−2∑xy≥0\sum_{}^{}{x^{2} - 2\sum_{}^{}{xy \geq 0}}

Ž ∑x2≥23∑xy\sum_{}^{}{x^{2} \geq \frac{2}{3}\sum_{}^{}{xy}}

Now, (x + y + z + w)2 = ∑x2+2∑xy≥23∑xy+2∑xy\sum_{}^{}{x^{2} + 2\sum_{}^{}{xy \geq \frac{2}{3}\sum_{}^{}{xy + 2\sum_{}^{}{xy}}}}

Ž (x + y + z + w)2 ³ 83∑xy\frac{8}{3}\sum_{}^{}{xy}

\ a2 ³83β\frac{8}{3}\beta.