\(P=\sqrt[]{x}+\dfrac{3}{\sqrt[]{x}-1}\left(x>1\right)\)

\(P=\sqrt[]{x}-1+\dfrac{3}{\sqrt[]{x}-1}+1\)

Áp dụng bất đẳng thức Cauchy cho 2 số \(\sqrt[]{x}-1;\dfrac{3}{\sqrt[]{x}-1}\) ta được :

\(\sqrt[]{x}-1+\dfrac{3}{\sqrt[]{x}-1}\ge2\sqrt[]{\sqrt[]{x}-1.\dfrac{3}{\sqrt[]{x}-1}}\)

\(\Rightarrow\sqrt[]{x}-1+\dfrac{3}{\sqrt[]{x}-1}\ge2\sqrt[]{3}\)

\(\Rightarrow P=\sqrt[]{x}-1+\dfrac{3}{\sqrt[]{x}-1}+1\ge2\sqrt[]{3}+1\)

\(\Rightarrow Min\left(P\right)=2\sqrt[]{3}+1\)

sorry mn cho e sửa lại đề ạ

tìm gtln của p ạ

ĐKXĐ : \(x>0\)

Áp dụng bất đẳng thức Cauchy cho 2 số dương \(\sqrt{x};\dfrac{4}{\sqrt{x}}\) ta có 

\(P=\sqrt{x}+\dfrac{4}{\sqrt{x}}\ge2\sqrt{\sqrt{x}.\dfrac{4}{\sqrt{x}}}=4\)

Dấu "=" xảy ra khi \(\sqrt{x}=\dfrac{4}{\sqrt{x}}\Leftrightarrow x=4\)

\(P=\sqrt[]{x}+\dfrac{4}{\sqrt[]{x}}\left(x>0\right)\)

\(P=\dfrac{x+4}{\sqrt[]{x}}=\dfrac{x+4}{\sqrt[]{x}}\)

Vì \(x>0;x+4>4\)

\(\Rightarrow P=\dfrac{x+4}{\sqrt[]{x}}>4\)

⇒ Không có giá trị nhỏ nhất

\(y=2+\dfrac{6}{x-3}\)

\(P=3x\left(2+\dfrac{6}{x-3}\right)+2x+2+\dfrac{6}{x-3}\)

\(P=8x+2+\dfrac{18x}{x-3}+\dfrac{6}{x-3}=8x+20+\dfrac{60}{x-3}\)

\(P=8\left(x-3\right)+\dfrac{60}{x-3}+44\ge2\sqrt{\dfrac{480\left(x-3\right)}{x-3}}+44=44+8\sqrt{30}\)

\(P_{min}=44+8\sqrt{30}\) khi \(8\left(x-3\right)=\dfrac{60}{x-3}\Leftrightarrow x=\dfrac{6+\sqrt{30}}{2}\)

ĐKXĐ: \(x\notin\left\{0;-9\right\}\)

Ta có: \(\dfrac{1}{x+9}-\dfrac{1}{x}=\dfrac{1}{5}+\dfrac{1}{4}\)

\(\Leftrightarrow\dfrac{20x}{20x\left(x+9\right)}-\dfrac{20\left(x+9\right)}{20x\left(x+9\right)}=\dfrac{4x\left(x+9\right)+5x\left(x+9\right)}{20x\left(x+9\right)}\)

Suy ra: \(4x^2+36x+5x^2+45x=20x-20x-180\)

\(\Leftrightarrow9x^2+81x+180=0\)

\(\Leftrightarrow x^2+9x+20=0\)

\(\Leftrightarrow x^2+4x+5x+20=0\)

\(\Leftrightarrow x\left(x+4\right)+5\left(x+4\right)=0\)

\(\Leftrightarrow\left(x+4\right)\left(x+5\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x+4=0\\x+5=0\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=-4\left(nhận\right)\\x=-5\left(nhận\right)\end{matrix}\right.\)

Vậy: S={-4;-5}

\(T=\dfrac{\sqrt{27}+3}{\sqrt{3}}=\dfrac{3\sqrt{3}+3}{\sqrt{3}}=\dfrac{3\left(\sqrt{3}+1\right)}{\sqrt{3}}=\sqrt{3}\left(\sqrt{3}+1\right)=3+\sqrt{3}\)

`T=\sqrt{27}+3/\sqrt{3}`

`T=3\sqrt{3}+\sqrt{3}`

`T=4\sqrt{3}`

Hướng làm:

Thấy cả tử mẫu cộng lại đều bằng 2021 → Cộng thêm 1 rồi quy đồng với mỗi phân thức

\(\dfrac{x+2}{2019}+1+\dfrac{x+3}{2018}+1=\dfrac{x+4}{2017}+1+\dfrac{x}{2021}+1\\ \Leftrightarrow\dfrac{x+2021}{2019}+\dfrac{x+2021}{2018}-\dfrac{x+2021}{2017}-\dfrac{x+2021}{2021}=0\\ \Leftrightarrow\left(x+2021\right)\left(\dfrac{1}{2019}+\dfrac{1}{2018}-\dfrac{1}{2017}-\dfrac{1}{2021}\right)=0\\ \Leftrightarrow x+2021=0\Leftrightarrow x=-2021\)

\(< =>\dfrac{x+2}{2019}+1+\dfrac{x+3}{2018}+1=\dfrac{x+4}{2017}+1+\dfrac{x}{2021}+1\)

\(< =>\dfrac{x+2+2019}{2019}+\dfrac{x+3+2018}{2018}=\dfrac{x+4+2017}{2017}+\dfrac{x+2021}{2021}\)

\(< =>\dfrac{x+2021}{2019}+\dfrac{x+2021}{2018}-\dfrac{x+2021}{2017}-\dfrac{x+2021}{2021}=0\)

\(< =>\left(x+2021\right)\left(\dfrac{1}{2019}+\dfrac{1}{2018}-\dfrac{1}{2017}-\dfrac{1}{2021}=\right)=0\)

\(< =>x+2021=0< =>x=-2021\)

Vậy....

1) \(A=36x^2+12x+1=\left(6x+1\right)^2\ge0\)

\(minA=0\Leftrightarrow x=-\dfrac{1}{6}\)

2) \(B=9x^2+6x+1=\left(3x+1\right)^2\ge0\)

\(minB=0\Leftrightarrow x=-\dfrac{1}{3}\)

4) \(D=x^2-4x+y^2-8y+6=\left(x-2\right)^2+\left(y-4\right)^2-14\ge-14\) 

\(minD=-14\Leftrightarrow\left\{{}\begin{matrix}x=2\\y=4\end{matrix}\right.\)

3) \(C=\left(x+1\right)\left(x-2\right)\left(x-3\right)\left(x-6\right)=\left(x^2-5x-6\right)\left(x^2-5x+6\right)=\left(x^2-5x\right)^2-36\ge-36\)

\(minC\Leftrightarrow\left[{}\begin{matrix}x=0\\x=5\end{matrix}\right.\)

5) \(E=\left(x-8\right)^2+\left(x+7\right)^2=2x^2-2x+113=2\left(x-\dfrac{1}{2}\right)^2+\dfrac{225}{2}\ge\dfrac{225}{2}\)

\(minE=\dfrac{225}{2}\Leftrightarrow x=\dfrac{1}{2}\)

e: Ta có: \(E=\left(x-8\right)^2+\left(x+7\right)^2\)

\(=x^2-16x+64+x^2+14x+49\)

\(=2x^2-2x+113\)

\(=2\left(x^2-x+\dfrac{113}{2}\right)\)

\(=2\left(x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{225}{4}\right)\)

\(=2\left(x-\dfrac{1}{2}\right)^2+\dfrac{225}{2}\ge\dfrac{225}{2}\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{1}{2}\)