Question

1) Chứng minh rằng: (a¹⁰ + b¹⁰)(a² + b²) \(\ge\) (a⁸ + b⁸)(a⁴ + b⁴)

2) Chứng minh rằng: a³(b² - c²) + b³(c² - b²) + c³(a² - b²) < 0

biết 0 < a < b < c

Answer 1

1) Xét hiệu:

\(\left(a^{10}+b^{10}\right)\left(a^2+b^2\right)-\left(a^8+b^8\right)\left(a^4+b^4\right)\)

\(=\left(a^{12}+a^{10}b^2+b^{10}a^1+b^{12}\right)-\left(a^{12}+a^8b^4+a^4b^8+b^{12}\right)\)

\(=a^{12}+a^{10}b^2+b^{10}a^2+b^{12}-a^{12}-a^8b^4-a^4b^8-b^{12}\)

\(=a^{10}b^2+b^{10}a^2-a^8b^4-a^4b^8\)

\(=a^2b^2\left(a^8+b^8-a^6b^2-a^2b^6\right)\)

\(=a^2b^2\left(a^2-b^2\right)\left(a^4+a^2b^2+b^4\right)\ge0\)

⇒ ĐPCM