Chủ đề này có 2 trả lời

[Thong Nhat]

1) Cho các số thực a, b thỏa mãn $3(a+b)\geq 2\left | ab+1 \right |$. Chứng minh: $9(a^3+b^3)\geq \left | a^3b^3+1 \right |$

2) Cho a, b > 0. Chứng minh: $\frac{1}{a^2}+\frac{1}{b^2}+\frac{4}{a^2+b^2}\geq \frac{32(a^2+b^2)}{(a+b)^4}$

3) Cho các số thực không âm a, b thỏa mãn $a^3+b^3=2$. Chứng minh: $3(a^4+b^4)+2a^4b^4\leq 8$

4) Cho số thực x, y thỏa mãn $y\geq 0; y(y+1)\leq(x+1)^2$. Chứng minh: $y(y-1)\leq x^2$

[DOTOANNANG]

$\lceil\,3\,\rfloor$

Đặt $\{\begin{array}{ll}a+ b= 2\,u\\ \\ ab= v\end{array}\,\rightarrow \,u^{2}\geqq v$

 

$2= a^{3}+ b^{3}= 2\,u\left [ 3\left ( u^{2}- v \right )+ u^{2} \right ]\geqq 2\,u\,u^{2}\Leftrightarrow u\leqq 1\Leftrightarrow 4\,u^{2}- 3\,v= 3\left ( \underbrace{u^{2}- v}_{\geqq 0} \right )+ u^{2}\geqq 1$

$8- 3\left ( a^{4}+ b^{4} \right )-2\,a^{4}b^{4}$ $= 8- 2\,v^{4}+ 6\,v^{2}- 6\left ( 2\,u^{2}- v \right )^{2}$ $= \left ( 1- v \right )\left [ \frac{148- 5\,\sqrt{10}}{27}+ \frac{1}{54}\left ( -6\,v+ \sqrt{10}- 2 \right )^{2}\left ( 3\,v+ \sqrt{10}+ 1 \right ) \right ]+ $ $+ 3\left ( 4\,u^{2}- 3\,v- 1 \right )\left [ 4\left ( u^{2}- v \right )+ 3\,v+ 1 \right ]\geqq 0$

[DOTOANNANG]

$\lceil\,4\,\rfloor$

$$\text{V}\left ( x,\,y \right )= x^{\,2}- y\left ( y- n \right )\geqq \text{V}\left ( x,\,\frac{n- \left | n \right |}{2} \right )\geqq \text{V}\left ( 0,\,\frac{n- \left | n \right |}{2} \right )\geqq 0$$

$\lceil\,2\,\rfloor$

$$\frac{1}{a^{\,2}}+ \frac{1}{b^{\,2}}+ \frac{4}{a^{\,2}+ b^{\,2}}- \frac{32\left ( a^{\,2}+ b^{\,2} \right )}{\left ( a+ b \right )^{\,4}}= \frac{\left ( a- b \right )^{\,4}\left ( a^{\,4}+ 8\,a^{\,3}b+ 6\,a^{\,2}b^{\,2}+ 8\,b^{\,3}a+ b^{\,4} \right )}{a^{\,2}b^{\,2}\left ( a+ b \right )^{\,4}\left ( a^{\,2}+ b^{\,2} \right )}\geqq 0$$