Giups mik vs

a) Ta có: \(\left(2x-1\right)^2\ge0\forall x\)

\(\Rightarrow-3\left(2x-1\right)^2\le0\forall x\)

\(\Rightarrow-3\left(2x-1\right)^2+5\le5\forall x\)

Dấu '=' xảy ra khi 2x-1=0

\(\Leftrightarrow2x=1\)

hay \(x=\dfrac{1}{2}\)

Vậy: Giá trị lớn nhất của biểu thức \(A=5-3\left(2x-1\right)^2\) là 5 khi \(x=\dfrac{1}{2}\)

1) \(x^2+2x+1=\left(x+2\right)\sqrt[]{x^2+1}\left(1\right)\)

\(\Leftrightarrow x^2+2x+1=x\sqrt[]{x^2+1}+2\sqrt[]{x^2+1}\left(x\ge-2\right)\)

\(\Leftrightarrow\left(x^2+2x+1\right)^2=\left(x\sqrt[]{x^2+1}+2\sqrt[]{x^2+1}\right)^2\)

\(\Leftrightarrow x^4+4x^2+1+4x^3+2x^2+4x=x^2\left(x^2+1\right)+4\left(x^2+1\right)+4x\left(x^2+1\right)\)

\(\Leftrightarrow x^4+4x^3+6x^2+4x+1=x^4+x^2+4x^2+4+4x^3+4\)

\(\Leftrightarrow x^4+4x^3+6x^2+4x+1=x^4+4x^3+5x^2+4x+4\)

\(\Leftrightarrow x^2=3\)

\(\Leftrightarrow x=\pm\sqrt[]{3}\left(Tm.x\ge-2\right)\)

Vậy nghiệm của phương trình \(\left(1\right)\) là \(x=\pm\sqrt[]{3}\)

2) \(P=\sqrt[]{x^2-2x+13}+4\sqrt[]{x-3}\)

Ta có : 

\(\sqrt[]{x^2-2x+13}=\sqrt[]{x^2-2x+1+12}=\sqrt[]{\left(x-1\right)^2+12}\ge\sqrt[]{12}=2\sqrt[]{3},\forall x\in R\)

\(4\sqrt[]{x-3}\ge0,\forall x\ge3\)

\(\Rightarrow P=\sqrt[]{x^2-2x+13}+4\sqrt[]{x-3}\ge\sqrt[]{4+12}+0=4\left(khi.x=3\right),\forall x\ge3\)

Vậy \(Min\left(P\right)=4\left(tại.x=3\right)\)

a: Ta có: \(\frac{x}{x^2-2x+1}+\frac{x}{1-x^2}\)

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

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

Ta có: \(A=\frac{x^2+x+1}{1-x^3}+\left(\frac{x}{x^2-2x+1}+\frac{x}{1-x^2}\right)\cdot\frac{x^2-1}{x^2+1}\)

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

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

\(\dfrac{a^2\cdot\sqrt[3]{a}\cdot\sqrt[5]{a^4}}{\sqrt[4]{a}}=\dfrac{a^2\cdot a^{\dfrac{1}{3}}\cdot a^{\dfrac{4}{5}}}{a^{\dfrac{1}{4}}}=\dfrac{a^{\dfrac{47}{15}}}{a^{\dfrac{1}{4}}}=a^{\dfrac{173}{60}}\)

\(\Rightarrow log_a\left(\dfrac{a^2\cdot\sqrt[3]{a}\cdot\sqrt[5]{a^4}}{\sqrt[4]{a}}\right)=log_a\left(a^{\dfrac{173}{60}}\right)=\dfrac{173}{60}\)

\(a^{2log_a\left(\dfrac{\sqrt{105}}{30}\right)}=a^{log_a\left(\dfrac{7}{60}\right)}=\dfrac{7}{60}\)

Vậy \(B=\dfrac{173}{60}+\dfrac{7}{60}=\dfrac{180}{60}=3\)