\(P=3x+2y+\frac{6}{x}+\frac{8}{y}\)

\(2P=6x+4y+\frac{12}{x}+\frac{16}{y}\)

\(=\left(3x+\frac{12}{x}\right)+\left(y+\frac{16}{y}\right)+3\left(x+y\right)\)

\(\ge2\sqrt{3x\cdot\frac{12}{x}}+2\sqrt{y\cdot\frac{16}{y}}+3\cdot6=12+8+18=38\)( bđt AM-GM và giả thiết x + y ≥ 6 )

=> P ≥ 19

Đẳng thức xảy ra <=> \(\hept{\begin{cases}3x=\frac{12}{x}\\y=\frac{16}{y}\\x+y=6\end{cases}}\Rightarrow\hept{\begin{cases}x=2\\y=4\end{cases}}\)

Vậy MinP = 19

Ta có: \(P=3x+2y+\frac{6}{x}+\frac{8}{y}=\left(\frac{3}{2}x+\frac{3}{2}y\right)+\left(\frac{3}{2}x+\frac{6}{x}\right)+\left(\frac{y}{2}+\frac{8}{y}\right)\)

Vì \(\frac{3}{2}x+\frac{3}{2}y=\frac{3}{2}\left(x+y\right)\ge\frac{3}{2}.6=9\)

\(\frac{3x}{2}+\frac{6}{x}\ge2\sqrt{\frac{3x}{2}.\frac{6}{x}}=6;\frac{y}{2}+\frac{8}{y}\ge2\sqrt{\frac{y}{2}.\frac{8}{y}}=4\)

\(\Rightarrow P\ge9+6+4=19\)

Dấu '=' xảy ra <=> \(\hept{\begin{cases}x+y=6\\\frac{3x}{2}=\frac{6}{x}\\\frac{y}{2}=\frac{8}{y}\end{cases}}\Leftrightarrow\hept{\begin{cases}x=2\\y=4\end{cases}}\)

Vậy GTNN của P là 19

1.

Do \(0\le a;b;c\le1\Rightarrow\left(1-a\right)\left(1-b\right)\left(1-c\right)\ge0\)

\(\Leftrightarrow1-abc-a-b-c+ab+bc+ca\ge0\)

\(\Leftrightarrow a+b+c-ab-bc-ca\le1-abc\le1\)

Mặt khác \(0\le a;b;c\le1\Rightarrow\left\{{}\begin{matrix}b^2\le b\\c^3\le c\end{matrix}\right.\)

\(\Rightarrow a+b^2+c^3-ab-bc-ca\le a+b+c-ab-bc-ca\le1\) (đpcm)

Dấu "=" xảy ra khi \(\left(a;b;c\right)=\left(0;0;1\right)\) và hoán vị

2.

\(P=3x+2y+\frac{6}{x}+\frac{8}{y}\)

\(P=\frac{3x}{2}+\frac{6}{x}+\frac{y}{2}+\frac{8}{y}+\frac{3x}{2}+\frac{3y}{2}\)

\(P=\left(\frac{3x}{2}+\frac{6}{x}\right)+\left(\frac{y}{2}+\frac{8}{y}\right)+\frac{3}{2}\left(x+y\right)\)

\(P\ge2\sqrt{\frac{18x}{2x}}+2\sqrt{\frac{8y}{2y}}+\frac{3}{2}.6=19\)

\(P_{min}=19\) khi \(\left\{{}\begin{matrix}x=2\\y=4\end{matrix}\right.\)

\(B=3x+2y+\frac{6}{x}+\frac{8}{y}\)

\(=\frac{3x}{2}+\frac{6}{x}+\frac{3x}{2}+\frac{y}{2}+\frac{8}{y}+\frac{3y}{2}\)

Áp dụng Cauchy ta được :

\(\frac{3x}{2}+\frac{6}{x}\ge2\sqrt{\frac{3x}{2}.\frac{6}{x}}=6\)

\(\frac{y}{2}+\frac{8}{y}\ge2\sqrt{\frac{8y}{2y}}=4\)

\(\Rightarrow B\ge6+4+\frac{3\left(x+y\right)}{2}\ge6+4+9=19\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}x+y=6\\\frac{y}{2}=\frac{8}{y}\\\frac{3x}{2}=\frac{6}{x}\end{cases}\Leftrightarrow x=2;y=4}\)

P = 3x + 2y + 6/x + 8/y 
P = (3x/2 + 6/x) + (3x/2 + 3y/2) + (y/2 + 8/y) 
Ta có 3x/2 + 6/x >= 2.căn (3x/2.6/x) = 6 
dấu = xảy ra khi 3x/2 = 6/x <=> x = 2 
3x/2 + 3y/2 = 3/2.(x+y) >= 3/2.6 = 9 
dấu = xảy ra khi x + y = 6 
y/2 + 8/y >= 2.căn (y/2.8/y) = 4 
Dấu = xảy ra khi y/2 = 8/y <=> y = 4 
Vậy P >= 6 + 9 + 4 <=> P > = 19 
Dấu = xảy ra khi x = 2 và y = 4 
=> P min = 19

sai roi ban phai dung ca x+y>=6 nua chu

Gợi ý nhé!  Tách rồi sử dụng Cauchy cho hai số ko âm

\(P=\left(3x+\frac{12}{x}\right)+\left(y+\frac{16}{y}\right)+2\left(x+y\right)\)

\(\ge2\sqrt{3.12}+2\sqrt{16}+2.6=32\)

"=" xảy ra <=> x=2; y=4

Ta có : \(P=5x+3y+\frac{12}{x}+\frac{16}{y}\) 

\(P=2\left(x+y\right)+\left(3x+\frac{12}{x}\right)+\left(y+\frac{16}{y}\right)\)  

Áp dụng BĐT Cô-si, ta có: \(3x+\frac{12}{x}\ge2\sqrt{\left(3.12\right)}=12\) 

\(y+\frac{16}{y}\ge2\sqrt{\left(1.16\right)}=8\) 

Ta có: \(x+y\ge6\) 

\(\Rightarrow2\left(x+y\right)\ge12\) 

\(\Rightarrow P\ge12+12+8=32\)

Dấu''='' xảy ra khi:

 \(3x=\frac{12}{x}\) , \(x+y=6\) , \(y=\frac{16}{y}\) 

\(\Rightarrow x=2,y=4\)

Vậy giá trị nhỏ nhất của P là 32 khi x = 2, y = 4

Đặt \(P=\frac{x^3}{y+z}+\frac{y+z}{4}\ge x;\frac{y^2}{z+x}+\frac{z+x}{4}\ge y;\frac{z^2}{x+y}+\frac{x+y}{4}\ge z\)

\(\Rightarrow P\ge x+y+x-\frac{x+y+z}{2}=\frac{x+y+z}{2}=\frac{4}{2}=2\)