(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.
+399 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.
