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Với a>0,b>0a>0,b>0 ta luôn có a+b≥2ab−−√a+b≥2ab
M = x2+y2xy=xy+yx=3xy+(x4y+yx)x2+y2xy=xy+yx=3xy+(x4y+yx)
Ta có: (x4y+yx)≥2x4y⋅yx−−−−−−√=1(x4y+yx)≥2x4y⋅yx=1
Mặt khác: x≥2yx≥2y ⇒3x4y≥32⇒3x4y≥32
Do đó M≥52M≥52 . Dâu ''='' xảy ra khi x=2yx=2y
Vậy giá trị nhỏ nhất của M là 5252 ⇔x=2y
\(P=\dfrac{1}{y}\left(\dfrac{1}{x}+\dfrac{1}{z}\right)\ge\dfrac{1}{y}.\dfrac{4}{x+z}=\dfrac{4}{y\left(x+z\right)}\ge\dfrac{4}{\dfrac{\left(y+x+z\right)^2}{4}}=4\)
\(P_{min}=4\) khi \(\left(x;y;z\right)=\left(\dfrac{1}{2};1;\dfrac{1}{2}\right)\)
\(x+y=4xy\Rightarrow\frac{x+y}{xy}=\frac{1}{x}+\frac{1}{y}=4\)
\(\frac{1}{x}+\frac{1}{y}>=\frac{4}{x+y}\Rightarrow4>=\frac{4}{x+y}\Rightarrow x+y>=1\)(bđt svacxo)
\(x^2+y^2>=\frac{\left(x+y\right)^2}{2};xy< =\frac{\left(x+y\right)^2}{4}\)
\(\Rightarrow P=x^2+y^2-xy>=\frac{\left(x+y\right)^2}{2}-\frac{\left(x+y\right)^2}{4}=\frac{\left(x+y\right)^2}{4}>=\frac{1^2}{4}=\frac{1}{4}\)
dấu = xảy ra khi \(x+y=1;x=y\Rightarrow x=y=\frac{1}{2}\left(tm\right)\)
vậy min P là \(\frac{1}{4}\)khi x=y=\(\frac{1}{2}\)
\(1\ge x+\dfrac{1}{y}\ge2\sqrt{\dfrac{x}{y}}\Rightarrow\dfrac{x}{y}\le\dfrac{1}{4}\)
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 \(\left(x;y\right)=\left(\dfrac{1}{2};2\right)\)
\(A=\frac{x}{x+1}+\frac{y}{y+1}+\frac{z}{z+1}\).Áp dụng BĐT Cauchy-Schwarz,ta có:
\(=\left(1-\frac{1}{x+1}\right)+\left(1-\frac{1}{y+1}\right)+\left(1-\frac{1}{z+1}\right)\)
\(=\left(1+1+1\right)-\left(\frac{1}{x+1}+\frac{1}{y+1}+\frac{1}{z+1}\right)\)
\(\ge3-\frac{9}{\left(x+y+z\right)+\left(1+1+1\right)}=\frac{3}{4}\)
Dấu "=" xảy ra khi x = y = z = 1/3
Vậy A min = 3/4 khi x=y=z=1/3
\(2x+2y\le1\Rightarrow x+y\le\frac{1}{2}\Rightarrow2\sqrt{xy}\le x+y\le\frac{1}{2}\)
\(\Rightarrow\sqrt{xy}\le\frac{1}{4}\Rightarrow xy\le\frac{1}{16}\Rightarrow\frac{1}{xy}\ge16\)
\(P=xy+\frac{1}{256xy}+\frac{255}{256xy}\ge2\sqrt{\frac{xy}{256xy}}+\frac{255}{256}.16=\frac{257}{16}\)
\(\Rightarrow P_{min}=\frac{257}{16}\) khi \(x=y=\frac{1}{4}\)