a: \(P=\dfrac{\sqrt{x}}{\sqrt{x}+3}+\dfrac{2\sqrt{x}}{\sqrt{x}-3}-\dfrac{3x+9}{x-9}\)
\(=\dfrac{\sqrt{x}\left(\sqrt{x}-3\right)+2\sqrt{x}\left(\sqrt{x}+3\right)-3x-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{x-3\sqrt{x}+2x+6\sqrt{x}-3x-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}\)
\(=\dfrac{3\sqrt{x}-9}{\left(\sqrt{x}+3\right)\left(\sqrt{x}-3\right)}=\dfrac{3}{\sqrt{x}+3}\)
b: \(\sqrt{x}+3>=3\forall x\) thỏa mãn ĐKXĐ
=>\(P=\dfrac{3}{\sqrt{x}+3}< =\dfrac{3}{3}=1\forall x\) thỏa mãn ĐKXĐ
Dấu '=' xảy ra khi x=0
c: \(Q=P+\dfrac{\sqrt{x}+15}{12}=\dfrac{3}{\sqrt{x}+3}+\dfrac{\sqrt{x}+15}{12}\)
\(=\dfrac{36+\left(\sqrt{x}+15\right)\left(\sqrt{x}+3\right)}{12\left(\sqrt{x}+3\right)}=\dfrac{x+18\sqrt{x}+81}{12\left(\sqrt{x}+3\right)}\)
\(=\dfrac{\left(\sqrt{x}+9\right)^2}{12\left(\sqrt{x}+3\right)}>=\dfrac{81}{12\cdot3}=\dfrac{81}{36}=\dfrac{9}{4}\)
Dấu '=' xảy ra khi x=0
