Question

CMR: \(\left(a+b\right)\left(a^3+b^3\right)\) ≤ \(2\left(a^4+b^4\right)\)

Answer 1

Lời giải:
Xét hiệu:

\(2(a^4+b^4)-(a+b)(a^3+b^3)=2(a^4+b^4)-(a^4+ab^3+a^3b+b^4)\)

\(=a^4+b^4-a^3b-ab^3=(a^4-a^3b)-(ab^3-b^4)\)

\(=a^3(a-b)-b^3(a-b)=(a^3-b^3)(a-b)=(a-b)(a^2+ab+b^2)(a-b)\)

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

Vì : \((a-b)^2\geq 0, \forall a,b\in\mathbb{R}\)

\(a^2+ab+b^2=(a+\frac{b}{2})^2+\frac{3}{4}b^2\geq 0, \forall a,b\in\mathbb{R}\)

\(\Rightarrow 2(a^4+b^4)-(a+b)(a^3+b^3)=(a-b)^2(a^2+ab+b^2)\geq 0\)

\(\Rightarrow 2(a^4+b^4)\geq (a+b)(a^3+b^3)\)

Ta có đpcm.