Question

CMR: \(8\left(a^4+b^4\right)\ge\left(a+b\right)^4\)

Answer 1

\(a.\) Ta có : \(\left(a-b\right)^2\) ≥ \(0\) ∀\(ab\)

⇔ \(a^2+b^2\text{ ≥}2ab\)

\(\text{⇔}a^4+2a^2b^2+b^4\text{≥}4a^2b^2\)

\(\text{⇔}a^4+b^4\text{≥}2a^2b^2\)

\(\text{⇔}a^4+b^4\text{≥ }\dfrac{1}{2}\left(a^2+b^2\right)^2\)

Cmtt , \(a^2+b^2\text{≥ }\dfrac{1}{2}\left(a+b\right)^2 \)

⇒ \(a^4+b^4\text{≥ }\dfrac{1}{8}\left(a+b\right)^4\)