Chuyển vế :
a^4 - a^3 + b^4 - b^3 + c^4 - c^3 >= 0
<=> a^3(a - 1) + b^3(b - 1) + c^3(c - 1) - (a - 1) - (b - 1) - (c - 1) >= 0 (a + b + c = 3)
<=> (a - 1)(a^3 - 1) + (b - 1)(b^3 - 1) + (c - 1)(c^3 - 1) >= 0
<=> (a - 1)^2(a^2 + a + 1) + (b - 1)^2(b^2 + b + 1) + (c -1)^2(c^2 + c + 1) >= 0 (*)
Dễ chứng minh được a^2 + a + 1 > 0
=> (*) đúng
=> ĐPCM
Cho a,b,c>0;a+b+c=3.cmr : a^3+b^3+c^3/a^4+b^4+c^4<hoac=1
a) Cho a3+b3+c3-3abc . CMR: a+b+c=0 ; a=b=c.
b) Cho a4+b4+c4+d4=4abcd và a,b,c,d >0 . CMR : a=b=c=d.
Cho a,b,c>0. CMR \(\frac{a^4}{a^3+b^3}+\frac{b^4}{b^3+c^3}+\frac{c^4}{c^3+a^3}\ge\frac{a+b+c}{2}\)
Cho a+b+c=3
CMR: \(a^4+b^4+c^4\ge a^3+b^3+c^3\)
cho a+b+c=3
cmr \(a^4+b^4+c^4>a^3+b^3+c^3\)
Cho a,b,c >0. CMR \(3\left(a^4+b^4+c^4\right)\ge\left(a+b+c\right)\left(a^3+b^3+c^3\right)\)
cho a,b,c,d > 0. CMR \(\frac{a^4}{a^3+2b^3}+\frac{b^4}{b^3+2c^3}+\frac{c^4}{c^3+2d^3}+\frac{d^4}{d^3+2a^3}\ge\frac{a+b+c+d}{3}\)
Cho a, b, c, d > 0. CMR \(\frac{a^4}{a^3+2b^3}+\frac{b^4}{a^3+2b^3}+\frac{c^4}{c^3+2d^3}+\frac{d^4}{d^3+2a^3}\ge\frac{a+b+c+d}{3}\)
Cho 3 số thực a,b,c thỏa mãn a+b+c=6.CMR: a4 + b4 + c4 ≥ 2*(a3 + b3 + c3)
