Question

Cho a,b,c thuộc R thỏa mãn a + b + c = 6
CMR: a^4 + b^4 + c^4 >= 2 (a^3 + b^3 + c^3)

Answer 1

Xét a⁴ - 2a³ \(\ge8a-16\)

<=> a⁴ -2a³ -8a +16\(\ge0\)

<=> (a⁴ - 2a³) - 8 (a-2) \(\ge0\)

<=> \(a^3\left(a-2\right)-8\left(a-2\right)\ge0\)

<=> \(\left(a-2\right)\left(a^3-8\right)\ge0\)

<=> \(\left(a-2\right)^2\left(a^2+2a+4\right)\ge0\) (luôn đúng)

Tương tự => \(\left\{{}\begin{matrix}b^4-2b^3\ge8b-16\\c^4-2c^3\ge8c-16\end{matrix}\right.\)

<=> \(a^4+b^4+c^4-2\left(a^3+b^3+c^3\right)\ge8\left(a+b+c\right)-48=0\)

<=> \(a^4+b^4+c^4\ge2\left(a^3+b^3+c^3\right)\)

Dấu "=" <=> a=b=c=2