neu de bai bai 1 la tinh x+y thi mik lam cho

đăng từng này thì ai làm cho 

We have \(P=\frac{x^4+2x^2+2}{x^2+1}\)

\(\Rightarrow P=\frac{x^4+2x^2+1+1}{x^2+1}\)

\(=\frac{\left(x^2+1\right)^2+1}{x^2+1}\)

\(=\left(x^2+1\right)+\frac{1}{x^2+1}\)

\(\ge2\sqrt{\frac{x^2+1}{x^2+1}}=2\)

(Dấu "="\(\Leftrightarrow x=0\))

Vậy \(P_{min}=2\Leftrightarrow x=0\)

Để \(\left(x^2-1\right)\left(x^2-16\right)< 0\) thì 

\(\hept{\begin{cases}x^2-1< 0\\x^2-16>0\end{cases}}\Leftrightarrow\hept{\begin{cases}x^2< 1\\x^2>16\end{cases}}\Leftrightarrow-4< x< -1\) hoặc \(\hept{\begin{cases}x< 1\\x>4\end{cases}}\) (loại)

Vậy \(-4< x< -1\)

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Câu hỏi của Phác Chí Mẫn - Toán lớp 9 | Học trực tuyến

Áp dụng BĐT \(a^2+b^2\ge\frac{\left(a+b\right)^2}{2}\)

\(\Rightarrow P\ge\frac{1}{2}\left(2x+\frac{1}{x}+2y+\frac{1}{y}\right)^2=\frac{1}{2}\left[2\left(x+y\right)+\frac{1}{x}+\frac{1}{y}\right]^2\)

\(\Rightarrow P\ge\frac{1}{2}\left[2\left(x+y\right)+\frac{4}{x+y}\right]^2=18\)

\(\Rightarrow P_{min}=18\) khi \(x=y=\frac{1}{2}\)

Đặt \(\left\{{}\begin{matrix}x=a^2\\y=b^2\end{matrix}\right.\) \(\Rightarrow12\ge\left(a^2+b^2\right)^3+4a^2b^2\ge8a^3b^3+4a^2b^2\)

\(\Rightarrow2a^3b^3+a^2b^2-3\le0\Rightarrow ab\le1\)

\(P=\frac{1}{1+a^2}+\frac{1}{1+b^2}+2018a^2b^2\le\frac{2}{1+ab}+2018a^2b^2\)

Ta sẽ chứng minh \(P\le2019\)

Thật vậy, đặt \(ab=t\Rightarrow0< t\le1\)

\(\frac{2}{1+t}+2018t^2\le2019\Leftrightarrow2+2018t^2\left(1+t\right)\le2019\left(1+t\right)\)

\(\Leftrightarrow2018t^3+2018t^2-2019t-2017\le0\)

\(\Leftrightarrow\left(t-1\right)\left(2018t^2+4036t+2017\right)\le0\) (luôn đúng)

(Do \(2018t^2+4036t+2017>0\) \(\forall t>0\) và \(t-1\le0\) \(\forall t\le1\))

\(\Rightarrow P_{max}=2019\) khi \(x=y=1\)

Ta coˊ :xy+x+1x​+yz+y+1y​+xz+z+1z​

=���+�+1+�����+��+�+����2��+���+��=xy+x+1x​+xyz+xy+xxy​+x2yz+xyz+xyxyz​

=���+�+1+����+�+1+1��+�+1(Vıˋ ���=1)=xy+x+1x​+xy+x+1xy​+xy+x+11​(Vıˋ xyz=1)

=�+��+1��+�+1=xy+x+1x+xy+1​

$=1$

\(P=\frac{2}{3xy}+\frac{3}{\sqrt{3\left(1+y\right)}}\ge\frac{2}{3y\left(3-y\right)}+\frac{6}{y+4}\)

\(\Rightarrow P\ge2\left(\frac{-9y^2+28y+4}{3\left(-y^3-y^2+12y\right)}\right)=2\left(\frac{2\left(-y^3-y^2+12y\right)+2y^3-7y^2+4y+4}{3\left(-y^3-y^2+12y\right)}\right)\)

\(P\ge2\left(\frac{2}{3}+\frac{\left(y-2\right)^2\left(2y+1\right)}{3y\left(3-y\right)\left(y+4\right)}\right)\ge\frac{4}{3}\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)

Đề gốc là \(P=\frac{x}{\sqrt{y}}+\frac{y}{\sqrt{z}}+\frac{z}{\sqrt{x}}\)

\(\frac{P}{4}=\frac{x}{2.2\sqrt{y}}+\frac{y}{2.2\sqrt{z}}+\frac{z}{2.2\sqrt{x}}\)

Áp dụng BĐT Côsi:

\(2.2.\sqrt{x}\le x+2^2=x+4\)

\(\Rightarrow\frac{P}{4}\ge\frac{x}{y+4}+\frac{y}{z+4}+\frac{z}{x+4}=\frac{x^2}{xy+4x}+\frac{y^2}{yz+4y}+\frac{z^2}{zx+4z}\)

\(\ge\frac{\left(x+y+z\right)^2}{xy+yz+zx+4\left(x+y+z\right)}\ge\frac{\left(x+y+z\right)^2}{\frac{1}{3}\left(x+y+z\right)^2+4\left(x+y+z\right)}=\frac{3\left(x+y+z\right)}{\left(x+y+z\right)+12}\)

\(=3-\frac{36}{x+y+z+12}\ge3-\frac{36}{12+12}=\frac{3}{2}\)

\(\Rightarrow P\ge6\)

Dấu bằng xảy ra khi \(x=y=z=4\)

+ Theo bđt cauchy :

\(\frac{1}{x^2+x}+\frac{x}{2}+\frac{x+1}{4}\ge3\sqrt[3]{\frac{1}{x\left(x+1\right)}\cdot\frac{x}{2}\cdot\frac{x+1}{4}}=\frac{3}{2}\)

Dấu "=" \(\Leftrightarrow\frac{1}{x\left(x+1\right)}=\frac{x}{2}=\frac{x+1}{4}\Leftrightarrow x=1\)

+ Tương tự :

\(\frac{1}{y^2+y}+\frac{y}{2}+\frac{y+1}{4}\ge\frac{3}{2}\) Dấu "=" <=> y = 1

\(\frac{1}{z^2+z}+\frac{z}{2}+\frac{z+1}{4}\ge\frac{3}{2}\) Dấu "=" <=> z = 1

Do đó : \(P+\frac{x+y+z}{2}+\frac{x+y+z+3}{4}\ge\frac{9}{2}\)

\(\Rightarrow P+\frac{3}{2}+\frac{3}{2}\ge\frac{9}{2}\) \(\Rightarrow P\ge\frac{3}{2}\)

Dấu "=" <=> x = y = z = 1