\(ĐKXĐ:x\ge0\)

\(\sqrt{x}+6\)luôn lớn hơn 4 thỏa mãn ĐKXĐ:

Vậy \(x\ge0\) là giá trị cần tìm

\(\sqrt{x}+6>4\Rightarrow\sqrt{x}>-2\)

Luôn đúng với mọi giá trị của x 

Ta có \(VT=\frac{a^2}{a\sqrt{b}}+\frac{b^2}{b\sqrt{c}}+\frac{c^2}{c\sqrt{a}}\ge\frac{\left(a+b+c\right)^2}{a\sqrt{b}+b\sqrt{c}+c\sqrt{a}}\)

Mà \(a\sqrt{b}\le\frac{a^2+b}{2},b\sqrt{c}\le\frac{b^2+c}{2},c\sqrt{a}\le\frac{c^2+a}{2}\)

=> \(VT\ge\frac{2\left(a+b+c\right)^2}{a^2+b^2+c^2+a+b+c}=\frac{2\left(a+b+c\right)^2}{3+a+b+c}\)

Lại có \(a+b+c\le\sqrt{3\left(a^2+b^2+c^2\right)}=3\)

=> \(VT\ge\frac{2\left(a+b+c\right)^2}{3+3}=\frac{\left(a+b+c\right)^2}{3}\ge ab+bc+ac\)(ĐPCM)

Dấu bằng xảy ra khi a=b=c=1

\(D=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2}{\sqrt{a}+\sqrt{b}}-\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{ab}}\)

\(=\frac{a-2\sqrt{ab}+b+4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{ab}}\)

\(=\frac{a+2\sqrt{ab}+b}{\sqrt{a}+\sqrt{b}}-\frac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{ab}}\)

\(=\frac{\left(\sqrt{a}+\sqrt{b}\right)^2}{\sqrt{a}+\sqrt{b}}-\left(\sqrt{a}-\sqrt{b}\right)\)

\(=\sqrt{a}+\sqrt{b}-\sqrt{a}+\sqrt{b}\)

\(=2\sqrt{b}\)

\(D=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2+4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}-\frac{a\sqrt{b}-b\sqrt{a}}{\sqrt{ab}}\)

\(D=\frac{\left(\sqrt{a}-\sqrt{b}\right)^2+4\sqrt{ab}}{\sqrt{a}+\sqrt{b}}-\frac{-b+\sqrt{a}.\sqrt{b}}{\sqrt{b}}\)

\(D=\frac{\left[\left(\sqrt{a}-\sqrt{b}\right)^2+4\sqrt{ab}\right].\sqrt{b}}{\left(\sqrt{a}+\sqrt{b}\right).\sqrt{b}}-\frac{\left(\sqrt{a}.\sqrt{b}-b\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{b}.\left(\sqrt{a}+\sqrt{b}\right)}\)

\(D=\frac{\left[\left(\sqrt{a}-\sqrt{b}\right)^2+4\sqrt{ab}\right]-\left(\sqrt{a}.\sqrt{b}-b\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{b}.\left(\sqrt{a}+\sqrt{b}\right)}\)

\(D=\frac{2b.\sqrt{a}+2b.\sqrt{b}}{\sqrt{b}.\left(\sqrt{a}+\sqrt{b}\right)}\)

\(D=\frac{2b.\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{b}\left(\sqrt{a}+\sqrt{b}\right)}\)

\(D=2\sqrt{b}\)

Ta có: B = 4x + 6y - x² - y² - 11 = -(x² - 4x + 4) - (y² - 6y + 9) + 2 = -(x - 2)² - (y - 3)² + 2

Ta luôn có: -(x - 2)² \(\le\)0 \(\forall\)x

          -(y - 3)² \(\le\)0 \(\forall\)y

=> -(x - 2)² - (y - 3)² + 2 \(\le\)2 \(\forall\)x; y

Dấu "=" xảy ra khi : \(\hept{\begin{cases}x-2=0\\y-3=0\end{cases}}\) <=> \(\hept{\begin{cases}x=2\\y=3\end{cases}}\)

Vậy Max của B = 2 tại x = 2 và y = 3

\(B=4x+6y-x^2-y^2-11.\)

\(=-\left[x^2-4x+y^2-6y+11\right]\)

\(=-\left[\left(x^2-4x+4\right)+\left(y^2-6y+9\right)-2\right]\)

\(=-\left(x-2\right)^2-\left(y-3\right)^2+2\)

\(B_{min}=2\Leftrightarrow\orbr{\begin{cases}x-2=0\\y-3=0\end{cases}\Rightarrow\orbr{\begin{cases}x=2\\y=3\end{cases}}}\)

\(E=\frac{2}{\sqrt{3}}+\frac{\sqrt{2}}{3}+\frac{2}{\sqrt{3}}.\left(\frac{5}{12}-\frac{1}{\sqrt{6}}\right)\)

\(E=\frac{2}{\sqrt{3}}+\frac{\sqrt{2}}{3}+\frac{5\sqrt{6}-12}{18\sqrt{2}}\)

\(E=\frac{36\sqrt{2}}{18\sqrt{6}}+\frac{12\sqrt{3}}{18\sqrt{6}}+\frac{\left(5\sqrt{6}-12\right).\sqrt{3}}{18\sqrt{3}}\)

\(E=\frac{36\sqrt{2}+12\sqrt{3}+\left(5\sqrt{6}-12\right).\sqrt{3}}{18\sqrt{6}}\)

\(E=\frac{51\sqrt{2}}{18\sqrt{6}}\)

\(E=\frac{17\sqrt{2}}{6\sqrt{6}}\)

\(E=\frac{17\sqrt{2}}{2.3\sqrt{2}.\sqrt{3}}\)

\(E=\frac{17}{\sqrt{2}.3\sqrt{2}.\sqrt{3}}\)

\(E=\frac{17}{6\sqrt{3}}\)

\(E=\frac{17\sqrt{3}}{18}\)

\(C=\frac{x-y}{\sqrt{x}-\sqrt{y}}\cdot\frac{\sqrt{x^3}-\sqrt{y^3}}{x+\sqrt{xy}+y}-2\sqrt{y}\)

\(=\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}\cdot\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{x+\sqrt{xy}+y}-2\sqrt{y}\)

\(=\left(\sqrt{x}+\sqrt{y}\right)\cdot\left(\sqrt{x}-\sqrt{y}\right)-2\sqrt{y}\)

\(=x-y-2\sqrt{y}\)

\(C=\frac{x-y}{\sqrt{x}-\sqrt{y}}.\frac{\sqrt{x^3}-\sqrt{y^3}}{x+\sqrt{xy}+y}-2\sqrt{y}.\)

\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}\)\(.\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{x+\sqrt{xy}+y}\)\(-2\sqrt{y}\)

\(=\left(\sqrt{x}+\sqrt{y}\right)\left(\sqrt{x}-\sqrt{y}\right)-2\sqrt{y}\)

\(=x-y-2\sqrt{y}\)