Cho 2 số thực x;y thỏa mãn x;y > bằng 1 và log_3 [(x+1)(y+1)]^(y+1)=9-(x-1)(y+1)

* Hướng dẫn giải

áp án B

Ta có:${\log }_3{\left[\left(x+1\right)\left(y+1\right)\right]}^{y+1}=9-\left(x-1\right)\left(y+1\right)\iff \left(y+1\right){\log }_3\left[\left(x+1\right)\left(y+1\right)\right]+\left(x-1\right)\left(y+1\right)=9$

$\iff \left(y+1\right){\log }_3\left[\left(c+1\right)\left(y+1\right)\right]+\left(x+1\right)\left(y+1\right)-2y=11$

$\iff \left(y+1\right)\left({\log }_3\left[\left(c+1\right)\left(y+1\right)\right]-2\right)=9-\left(x+1\right)\left(y+1\right)\left(*\right)$

 Nếu  $\left(x+1\right)\left(y+1\right)>9\Rightarrow VT\left(*\right)>0;VP\left(*\right)<0$

Ngược lại nếu  $\left(x+1\right)\left(y+1\right)<9\Rightarrow VT\left(*\right)<0;VP\left(*\right)>0$

Do đó  $\left(*\right)\iff \left(x+1\right)\left(y+1\right)=9\iff xy+x+y=8$

Khi đó  $P={\left(x+y\right)}^3-3xy\left(x+y\right)-57\left(x+y\right)={\left(x+y\right)}^3-3\left(8-x-y\right)\left(x+y\right)-57\left(x+y\right)$

Đặt  $t=\left(x+y\right)\ge 2\Rightarrow f\left(t\right)=t^3-3\left(8-t\right)t-57t=t^3+3t^2-81t$

$\Rightarrow f\text{'}\left(t\right)=3t^2+6t-81=0\Rightarrow t=-1+2\sqrt{7}\Rightarrow P_{\min }=f\left(-1+2\sqrt{7}\right)=83-112\sqrt{7}\Rightarrow a+b=-29$