Ta biến đổi: \(4\left(a^3+b^3\right)-\left(a+b\right)^3+4\left(b^3+c^3\right)-\left(b+c\right)^3+4\left(c^3+a^3\right)-\left(c+a\right)^3\ge0\)
Xét: \(4\left(a^3+b^3\right)-\left(a+b\right)^3=\left(a+b\right)\left[4\left(a^2-ab+b^2\right)-\left(a+b\right)^2\right]\)
\(=3\left(x+b\right)\left(a-b\right)^2\ge0\)
Tương tự với: \(4\left(b^3+c^3\right)-\left(b+c\right)^3\) và \(4\left(c^3+a^3\right)-\left(c+a\right)^3\)
Ta suy ra đpcm.
Đẳng thức xảy ra \(\Leftrightarrow a=b=c\)
CMR: \(3\left(a^8+b^8+c^8\right)\ge\left(a^3+b^3+c^3\right)\left(a^5+b^5+c^5\right)\)
Cho a, b, c > 0 . CMR:
\(\frac{1}{a+b+c}\ge\frac{a^3}{\left(2a^2+b^2\right)\left(2a^2+c^2\right)}+\frac{b^3}{\left(2b^2+c^2\right)\left(2b^2+a^2\right)}+\frac{c^3}{\left(2c^2+a^2\right)\left(2c^2+a^2\right)}\)
CMR: \(\left(a^3+b^3+c^3\right)\ge a^3+b^3+c^3+24abc\) với a, b, c ≥ 0
Cho a,b,c khác 0 thỏa mãn \(\left(1+\dfrac{b}{a}\right)\left(1+\dfrac{c}{b}\right)\left(1+\dfrac{a}{c}\right)=8\)
CMR \(\dfrac{a}{a+b}+\dfrac{b}{b+c}+\dfrac{c}{c+a}=\dfrac{3}{4}+\dfrac{ab}{\left(a+b\right)\left(b+c\right)}+\dfrac{bc}{\left(b+c\right)\left(c+a\right)}+\dfrac{ca}{\left(c+a\right)\left(a+b\right)}\)
CMR: \(2\left(a^3+b^3\right)\) ≥ \(\left(a+b\right)\left(a^2+b^2\right)\) với a, b > 0
1. Chứng minh rằng: \(3\left(a^8+b^8+c^8\right)\ge\left(a^3+b^3+c^3\right)\left(a^5+b^5+c^5\right)\)
2. Cho a+b=2. Chứng minh rằng: \(a^8+b^8\ge a^7+b^7\)
CMR: \(4\left(a^3+b^3\right)\) ≥ \(\left(a+b\right)^3\) với a, b > 0
CM CÁC BẤT ĐẲNG THỨC SAU
A) \(2\left(A^2+B^2\right)\ge\left(A+B\right)^2\ge2\left(AB+BA\right)\)
B) \(3\left(A^2+B^2+C^2\right)\ge\left(A+B+C\right)^2\ge3\left(AB+BC+CA\right)\)
Cho a, b, c là ba số dương thoả mãn abc = 1. Chứng minh rằng: \(\dfrac{1}{a^3\left(b+c\right)}+\dfrac{1}{b^3\left(c+a\right)}+\dfrac{1}{c^3\left(a+b\right)}\ge\dfrac{3}{2}\)
