(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.
+27 (a) Proof: Let x = a^2 + b^2, y = c^4 + d^4, and z = e^4 + f^4. By the AM-GM inequality, we have xyz >= 8xyz. Expanding this, we get (a^2 + b^2)^2 (c^4 + d^4)(e^4 + f^4) >= 8(a^2 + b^2)(c^4 + d^4)(e^4 + f^4). Simplifying, we get (a^2 + b^2)^2 (c^4 + d^4)(e^4 + f^4) >= 8abcdef. Since 8abcdef >= 32abcdef, we have (a^2 + b^2)^2 (c^4 + d^4)(e^4 + f^4) >= 32abcdef. (b) Proof: Let x = a^2 + b^2, y = c^2 + d^2, and z = e^2 + f^2. By the AM-GM inequality, we have xyz >= 4xyz. Expanding this, we get (a^2 + b^2)(c^2 + d^2)(e^2 + f^2) >= 4(a^2 + b^2)(c^2 + d^2)(e^2 + f^2). Simplifying, we get (a^2 + b^2)(c^2 + d^2)(e^2 + f^2) >= 8abcdef.
