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(a) Prove that if a, b, c, d, e, f are nonnegative real numbers, then (a^2 + b^2)^2 (c^4 + d^4)(e^4 + f^4) >= 32abcdef.

(b) Prove that if a, b, c, d, e, f are nonnegative real numbers, then (a^2 + b^2)(c^2 + d^2)(e^2 + f^2) >= 8abcdef.

 Dec 29, 2022
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Part B)
We can prove this by using the trivial inequality:

\((a-b)^2>=0\)

We expand

\(a^2-2ab+b^2>=0\)

Rearrange

\(a^2+b^2>=2ab\)

Doing this with all the variables gets us \(c^2+d^2>=2cd\) and \(e^2+f^2>=2ef\)

Multiplying these inequalities gets us, \((a^2+b^2)(c^2+d^2)(e^2+f^2)>=8abcdef\)

Note: We can multiply these inequalities because they are positive, and they have the same signs.

 Dec 31, 2022

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