(1)柯西不等式
\((\displaystyle \frac{a^2}{2a+b}+\frac{b^2}{2b+c}+\frac{c^2}{2c+a})\)\((3a+3b+3c)\geq (a+b+c)^2\)


\(\displaystyle \frac{a^2}{2a+b}+\frac{b^2}{2b+c}+\frac{c^2}{2c+a}\geq \frac{a+b+c}{3}\)

(2)原式即証\(\displaystyle (1+r_1^3)(1+r_2^3)(1+r_3^3)\geq (1+r_1r_2r_3)^3\)
其中\(\displaystyle r_1=\frac{a_2}{a_1},r_2=\frac{b_2}{b_1},r_3=\frac{c_2}{c_1}\)

\(\displaystyle (1+r_1^3)(1+r_2^3)(1+r_3^3)=1+(r_1^3+r_2^3+r_3^3)+[(r_1r_2)^3+(r_2r_3)^3+(r_3r_1)^3]+(r_1r_2r_3)^3\geq 1+3r_1r_2r_3+3(r_1r_2r_3)^2+(r_1r_2r_3)^3=(1+r_1r_2r_3)^3\)得証

[ 本帖最後由 satsuki931000 於 2021-5-7 21:20 編輯 ]

以下幾題想對個答案
8(4) \(\displaystyle \frac{1}{2}(e^{-x}sinx - e^{-x}cosx)+C\)

8(2) \(\displaystyle [sin(x^2+1)]^x [ln \ sin(x^2+1)+2x^2\ cot(x^2+1)]+C\)

8(3) \(\displaystyle \frac{2}{3}\sqrt{1+x^3}(\frac{x^3}{3}-\frac{2}{3})+C\)

筆誤打錯 謝謝您的提醒