Question

\(\frac{3}{x}+\frac{2}{y}+\frac{6}{2x+3y}\left(x>0;y>0;xy=6\right) Tìm-GTNN\)

Answer 1

tìm GTNN nhé.giúp mình.mik cám ơn nhiều

Answer 2

M min khi và chỉ khi x=3;y=2

Answer 3

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

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

\(=\frac{3.\left(3y+2x\right)}{4.6}+\frac{3y+2x}{24}+\frac{6}{2x+3y}\)

\(=\frac{3.\left(3y+2x\right)}{24}+\frac{3y+2x}{24}+\frac{6}{2x+3y}\)

Áp dụng BĐT AM-GM ta có:

\(\frac{3.\left(3y+2x\right)}{24}+\frac{3y+2x}{24}+\frac{6}{2x+3y}\ge\frac{3.2.\sqrt{3y.2x}}{24}+2.\sqrt{\frac{\left(3y+2x\right)}{24}.\frac{6}{\left(2x+3y\right)}}=\frac{6.\sqrt{6.6}}{24}+2.\sqrt{\frac{1}{4}}=\frac{3}{2}+1=2,5\)

\(\Rightarrow\frac{3}{x}+\frac{2}{y}+\frac{6}{2x+3y}\ge2,5\)

Dấu '" = " xảy ra \(\Leftrightarrow\hept{\begin{cases}\frac{3y+2x}{24}=\frac{6}{3y+2x}\\3y=2x\end{cases}}\Leftrightarrow\hept{\begin{cases}\left(3y+2x\right)^2=144\\\frac{y}{2}=\frac{x}{3}\end{cases}}\Leftrightarrow\hept{\begin{cases}3y+2x=12\left(v\text{ì}x,y>0\right)\\\frac{3y}{6}=\frac{2x}{6}\end{cases}}\)

Áp dụng tính chất dãy tỉ số bằng nhau ta có:

\(\frac{3y}{6}=\frac{2x}{6}=\frac{3y+2x}{6+6}=\frac{12}{12}=1\)

\(\Rightarrow\hept{\begin{cases}\frac{3y}{6}=1\\\frac{2x}{6}=1\end{cases}}\Leftrightarrow\hept{\begin{cases}3y=6\\2x=6\end{cases}}\Leftrightarrow\hept{\begin{cases}y=2\\x=3\end{cases}}\)

Vậy GTNN của \(\frac{3}{x}+\frac{2}{y}+\frac{6}{2x+3y}=2,5\Leftrightarrow x=3;y=2\)

Tham khảo nhé~