ý C phải ko

ý c phải ko

2 ) 

\(\frac{a+b}{2}.\frac{a^2+b^2}{2}\le\frac{a^3+b^3}{2}\)

\(\Leftrightarrow\frac{a^3+b^3+a^2b+ab^2}{4}\le\frac{a^3+b^3}{2}\)

\(\Leftrightarrow a^3+b^3+a^2b+ab^2\le2a^3+2b^3\)

\(\Leftrightarrow a^2\left(a-b\right)-b^2\left(a-b\right)\ge0\)

\(\Leftrightarrow\left(a-b\right)^2\left(a+b\right)\ge0\) ( luôn đúng ) 

\(\hept{\begin{cases}\left(a-b\right)^2\ge0\\a>0;b>0\Rightarrow a+b>0\end{cases}}\)

\(ĐKXĐ:0\le x\le1\)

Đặt \(\hept{\begin{cases}\sqrt[4]{x}=a\\\sqrt[4]{1-x}=b\\\sqrt[4]{\frac{1}{2}}=c\end{cases}}\left(a,b,c\ge0\right)\)

Ta có hpt : 

\(\hept{\begin{cases}a+a^2+b+b^2=2c+2c^2\\a^4+b^4=2=2c^4\end{cases}\left(^∗\right)}\)

Áp dụng BĐT : 

\(a^2+b^2\le\sqrt{2\left(a^4+b^4\right)}=\sqrt{2.2c^4}=2c^2\left(c>0\right)\left(1\right)\)

\(a+b\le\sqrt{2\left(a^2+b^2\right)}\le\sqrt{2.2c^2}=2c\left(2\right)\)

\(\left(1\right)+\left(2\right)\) vế theo vế \(\Rightarrow a^2+b^2+a+b\le2c^2+2c\)

Để dấu " = " ở (* ) xảy ra 

\(\Rightarrow a=b\Rightarrow a^4=b^4\Rightarrow x=1-x\Rightarrow x=\frac{1}{2}\left(TMĐKXĐ\right)\)

Câu 1 là \(\left(8x-4\right)\sqrt{x}-1\) hay là \(\left(8x-4\right)\sqrt{x-1}\)?

Câu 1:ĐK \(x\ge\frac{1}{2}\)

\(4x^2+\left(8x-4\right)\sqrt{x}-1=3x+2\sqrt{2x^2+5x-3}\)

<=> \(\left(4x^2-3x-1\right)+4\left(2x-1\right)\sqrt{x}-2\sqrt{\left(2x-1\right)\left(x+3\right)}\)

<=> \(\left(x-1\right)\left(4x+1\right)+2\sqrt{2x-1}\left(2\sqrt{x\left(2x-1\right)}-\sqrt{x+3}\right)=0\)

<=> \(\left(x-1\right)\left(4x+1\right)+2\sqrt{2x-1}.\frac{8x^2-4x-x-3}{2\sqrt{x\left(2x-1\right)}+\sqrt{x+3}}=0\)

<=>\(\left(x-1\right)\left(4x+1\right)+2\sqrt{2x-1}.\frac{\left(x-1\right)\left(8x+3\right)}{2\sqrt{x\left(2x-1\right)}+\sqrt{x+3}}=0\)

<=> \(\left(x-1\right)\left(4x+1+2\sqrt{2x-1}.\frac{8x+3}{2\sqrt{x\left(2x-1\right)}+\sqrt{x+3}}\right)=0\)

Với \(x\ge\frac{1}{2}\)thì \(4x+1+2\sqrt{2x-1}.\frac{8x-3}{2\sqrt{x\left(2x-1\right)}+\sqrt{x+3}}>0\)

=> \(x=1\)(TM ĐKXĐ)

Vậy x=1