Lời giải:
Áp dụng BĐT AM-GM:
\(a^3+a\geq 2a^2; b^3+b\geq 2b^2; c^3+c\geq 2c^2\)
\(\Rightarrow A=\frac{a}{a^3+a+1}+\frac{b}{b^3+b+1}+\frac{c}{c^3+c+1}\leq \frac{a}{2a^2+1}+\frac{b}{2b^2+1}+\frac{c}{2c^2+1}\)
\(\leq \frac{a}{a^2+2a}+\frac{b}{b^2+2b}+\frac{c}{c^2+2c}\)
hay \(A\leq \frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}(1)\)
Vì $abc=1$ nên đặt \((a,b,c)=(\frac{x}{y}, \frac{y}{z}, \frac{z}{x})(x,y,z>0)\)
Khi đó:
\(\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}=\frac{y}{x+2y}+\frac{z}{y+2z}+\frac{x}{z+2x}=\frac{1}{2}(1-\frac{x}{x+2y}+1-\frac{y}{y+2z}+1-\frac{z}{z+2x})\)
\(=\frac{3}{2}-\frac{1}{2}(\frac{x^2}{x^2+2xy}+\frac{y^2}{y^2+2zy}+\frac{z^2}{z^2+2xz})\)
\(\leq \frac{3}{2}-\frac{1}{2}.\frac{(x+y+z)^2}{x^2+2xy+y^2+2zy+z^2+2xz}=\frac{3}{2}-\frac{1}{2}.\frac{(x+y+z)^2}{(x+y+z)^2}=1(2)\) (theo BĐT Cauchy-Schwarz)
Từ \((1);(2)\Rightarrow A\leq 1\) (đpcm)
Dấu "=" xảy ra khi $a=b=c=1$
bai n ay la bai o dau ma dau cung thay nhi
\(\left(a^{\dfrac{1}{3}};b^{\dfrac{1}{3}};c^{\dfrac{1}{3}}\right)->\left(x;y;z>0\right)\left(xyz=1\right)\)\(\RightarrowΣ\dfrac{x^3}{x^9+x^3+1}\le1\)
\(\dfrac{x^3}{x^9+x^3+1}\le\dfrac{x^2+1}{2\left(x^4+x^2+1\right)}\)
\(\Leftrightarrow-\dfrac{\left(x-1\right)^2\left(x^9+2x^8+4x^7+6x^6+6x^5+6x^4+5x^3+4x^2+2x+1\right)}{2\left(x^2-x+1\right)\left(x^2+x+1\right)\left(x^9+x^3+\right)}\le0\)
\(\Rightarrow VT\le\dfrac{1}{2}\cdot2=1=VP\)
a=b=c=x=y=z=1
