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\(2x+y=\frac{x}{2}+\frac{x}{2}+\frac{x}{2}+\frac{x}{2}+y\ge5\sqrt[5]{\frac{x^4y}{16}}\)
\(5x^2+5y^2=\frac{5}{4}x^2+\frac{5}{4}x^2+\frac{5}{4}x^2+\frac{5}{4}x^2+5y^2\ge5\sqrt[5]{\frac{5^5}{4^4}x^8y^2}=5^2.\sqrt[5]{\frac{1}{4^4}}.\left(\sqrt[5]{x^4y}\right)^2\)
\(\Rightarrow\sqrt{5x^2+5y^2}\ge5.\sqrt[5]{\frac{1}{2^4}}.\sqrt[5]{x^4y}\)
\(10=2x+y+\sqrt{5x^2+5y^2}\ge10.\sqrt[5]{\frac{1}{16}}\sqrt[5]{x^4y}\)
\(\Rightarrow\sqrt[5]{x^4y}\le\sqrt[5]{16}\)\(\Rightarrow x^4y\le16\)
Theo mk nghĩ đề đúng thì chắc cách giải như zầy
\(\Rightarrow\hept{\begin{cases}x+\sqrt{1+x^2}=\frac{1}{y+\sqrt{1+y^2}}\\y+\sqrt{1+y^2}=\frac{1}{x+\sqrt{1+x^2}}\end{cases}}\)\(\Leftrightarrow\hept{\begin{cases}x+\sqrt{1+x^2}-\sqrt{1+y^2}+y=0\\y+\sqrt{1+y^2}-\sqrt{1+x^2}+x=0\end{cases}}\)
\(\Leftrightarrow2x+2y=0\Leftrightarrow x+y=0\)
\(P=2x-3\sqrt{xy}+y=2x-3\sqrt{xy}+y+\left(-x-\sqrt{xy}+4y-4\sqrt{y}+16\right)\)
\(=x-4\sqrt{xy}+5y-4\sqrt{y}+16\)
\(=\left(\sqrt{x}-2\sqrt{y}\right)^2+\left(\sqrt{y}-2\right)^2+12\ge12\)
Dấu \(=\)xảy ra khi \(\hept{\begin{cases}\sqrt{x}=2\sqrt{y}\\\sqrt{y}-2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=16\\y=4\end{cases}}\).
Với \(x=16,y=4\)thỏa mãn giả thiết.
Vậy \(minP=12\).
\(\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1\)
Nhân hai vế của đẳng thức với: \(\sqrt{x^2+1-x}\)
Ta được: \(\left(x+\sqrt{x^2+1}\right)\left(\sqrt{x^2+1}-x\right)\left(y+\sqrt{y^2+1}\right)=\sqrt{x^2+1}-x\)
\(\Leftrightarrow y+\sqrt{y^2+1}=\sqrt{x^2+1}-x\)
\(\Leftrightarrow x+y=\sqrt{x^2+1}-\sqrt{y^2+1}\left(1\right)\)
Mặt khác ta có: \(\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1\)
Nhân hai vế của đẳng thức với: \(\sqrt{y^2+1}-y\)
Ta được: \(\left(x+\sqrt{x^2+1}\right)\left(\sqrt{y^2+1}-y\right)\left(\sqrt{y^2+1}+y\right)=\sqrt{y^2+1}-y\)
\(\Leftrightarrow x+\sqrt{x^2+1}=\sqrt{y^2+1}-y\)
\(\Leftrightarrow x+y=\sqrt{y^2+1}-\sqrt{x^2+1}\left(2\right)\)
Từ: \(\left(1\right)\left(2\right)\Rightarrow x+y=0\left(đpcm\right)\)
\(a^2+b^2=\left(a+b-c\right)^2=a^2+\left(b-c\right)^2+2a\left(b-c\right)=b^2+\left(a-c\right)^2+2b\left(a-c\right)\)
\(\Rightarrow\left\{{}\begin{matrix}b^2=\left(b-c\right)^2+2a\left(b-c\right)\\a^2=\left(a-c\right)^2+2b\left(a-c\right)\end{matrix}\right.\)
\(\Rightarrow\dfrac{a^2+\left(a-c\right)^2}{b^2+\left(b-c\right)^2}=\dfrac{\left(a-c\right)^2+2b\left(a-c\right)+\left(a-c\right)^2}{\left(b-c\right)^2+2a\left(b-c\right)+\left(b-c\right)^2}\)
\(=\dfrac{\left(a-c\right)\left(a+b-c\right)}{\left(b-c\right)\left(b+a-c\right)}=\dfrac{a-c}{b-c}\) (đpcm)
\(\sqrt{x\left(1-y\right)\left(1-z\right)}=\sqrt{x\left(yz-y-z+1\right)}=\sqrt{x\left(yz-y-z+x+y+z+2\sqrt{xyz}\right)}\)
\(=\sqrt{x\left(yz+x+2\sqrt{xyz}\right)}=\sqrt{x^2+2x\sqrt{xyz}+xyz}=\sqrt{\left(x+\sqrt{xyz}\right)^2}\)
\(=x+\sqrt{xyz}\)
Tương tự: \(\sqrt{y\left(1-x\right)\left(1-z\right)}=y+\sqrt{xyz}\) ; \(\sqrt{z\left(1-x\right)\left(1-y\right)}=z+\sqrt{xyz}\)
\(\Rightarrow VT=x+y+z+3\sqrt{xyz}=1-2\sqrt{xyz}+3\sqrt{xyz}=1+\sqrt{xyz}\) (đpcm)
Áp dụng BĐT Bunhiacopski:
Đặt \(A=x\sqrt{16-y}+\sqrt{y\left(16-x^2\right)}\)
\(\Leftrightarrow A^2=\left[x\sqrt{16-y}+\sqrt{y\left(16-x^2\right)}\right]^2\le\left(x^2+16-x^2\right)\left(16-y+y\right)\\ \Leftrightarrow A^2\le16\cdot16=256\\ \Leftrightarrow A\le16\\ A_{max}=16\Leftrightarrow\dfrac{x^2}{16-x^2}=\dfrac{16-y}{y}\Leftrightarrow x^2y=256-16y-16x^2+x^2y\\ \Leftrightarrow16x^2+16y-256=0\\ \Leftrightarrow x^2+y-16=0\\ \Leftrightarrow x^2=16-y\Leftrightarrow x=\sqrt{16-y}\)