Ta có

ĐÁP ÁN C

\(P=x^2-3x+\dfrac{1}{2x}+\dfrac{7}{4}+\dfrac{1}{4}\)

\(P=\dfrac{4x^3-12x^2+7x+2}{4x}+\dfrac{1}{4}=\dfrac{\left(x-2\right)\left(4x^2-4x-1\right)}{4x}+\dfrac{1}{4}\)

\(P=\dfrac{\left(x-2\right)\left[4x\left(x-2\right)+\dfrac{1}{2}\left(x-2\right)+\dfrac{7x}{2}\right]}{4x}+\dfrac{1}{4}\ge\dfrac{1}{4}\)

\(P_{min}=\dfrac{1}{4}\) khi \(x=2\)

\(P=x^2-3x+\dfrac{1}{2x}+2\)

\(P=x^2-4x+4+x+\dfrac{4}{x}-\dfrac{7}{2x}-2\)

\(P=\left(x-2\right)^2+x+\dfrac{4}{x}-\dfrac{7}{2x}-2\)

Áp dụng bđt cosi và bđt x \(\ge\)2

Ta có: P \(\ge0+2\sqrt{x\cdot\dfrac{4}{x}}-\dfrac{7}{2.2}-2=\dfrac{1}{4}\)

Dấu "=" xảy ra <=> x = 2

Vậy MinP = 1/4 <=> x = 2

\(1\ge x+\dfrac{1}{y}\ge2\sqrt{\dfrac{x}{y}}\Rightarrow\dfrac{x}{y}\le\dfrac{1}{4}\)

Đặt \(\dfrac{x}{y}=a\Rightarrow0< a\le\dfrac{1}{4}\)

\(P=\dfrac{\left(\dfrac{x}{y}\right)^2-\dfrac{2x}{y}+2}{\dfrac{x}{y}+1}=\dfrac{a^2-2a+2}{a+1}=\dfrac{4a^2-8a+8}{4\left(a+1\right)}=\dfrac{4a^2-13a+3+5\left(a+1\right)}{4\left(a+1\right)}\)

\(P=\dfrac{5}{4}+\dfrac{\left(1-4a\right)\left(3-a\right)}{4\left(a+1\right)}\ge\dfrac{5}{4}\)

Dấu "=" xảy ra khi \(a=\dfrac{1}{4}\) hay \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)

\(M=\dfrac{2x+y}{xy}+\dfrac{3}{2x+y}=\dfrac{2x+y}{2}+\dfrac{3}{2x+y}=\dfrac{3\left(2x+y\right)}{16}+\dfrac{3}{2x+y}+\dfrac{5}{16}\left(2x+y\right)\ge2\sqrt{\dfrac{3}{16}.3}+\dfrac{5}{16}.2\sqrt{2xy}=\dfrac{3}{2}+\dfrac{5}{4}=\dfrac{11}{4}\).

Đẳng thức xảy ra khi x = 1; y = 2.

\(M=\dfrac{2x+y}{xy}+\dfrac{3}{2x+y}=\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\)

\(M=\dfrac{3\left(2x+y\right)}{16}+\dfrac{3}{2x+y}+\dfrac{5\left(2x+y\right)}{16}\ge2\sqrt{\dfrac{9\left(2x+y\right)}{16\left(2x+y\right)}}+\dfrac{5}{16}.2\sqrt{2xy}=\dfrac{11}{4}\)

Dấu "=" xảy ra khi \(\left(x;y\right)=\left(1;2\right)\)

Ta có:

\(M=\dfrac{2x+y}{xx}+\dfrac{3}{2x+y}=\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\)

\(=\left(\dfrac{3}{8}\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\right)+\dfrac{5}{8}\dfrac{2x+y}{2}\)

Có: \(\dfrac{3}{8}\dfrac{2x+y}{2}+\dfrac{3}{2x+y}\ge2\sqrt{\dfrac{3}{8}\dfrac{2x+y}{2}\dfrac{3}{2x+y}}=\dfrac{3}{2}\)

Dấu '=' xảy ra \(\Leftrightarrow\dfrac{3}{8}\dfrac{2x+y}{2}=\dfrac{3}{2x+y}\)

Có: \(\dfrac{5}{8}\dfrac{2x+y}{2}\ge\dfrac{5}{8}\sqrt{2xy}=\dfrac{5}{4}\)

Dấu '=' xảy ra \(\Leftrightarrow2x=y,xy=2\)

\(\Rightarrow M\ge\dfrac{3}{2}+\dfrac{5}{4}=\dfrac{11}{4}\)

Dấu '=' xảy ra \(\Leftrightarrow x=1,y=2\)

Vậy GTNN của M là \(\dfrac{11}{4}\Leftrightarrow x=1,y=2\)

\(M=\dfrac{2x+y}{xy}\)