F=\(\sqrt{x^2+2019}\)

=>\(F^2=x^2+2019 =>x^2+2019\)≥2019

=> \(F^2 \)_min=2019=>F _min=\(\sqrt{2019}\)<=>x=0

G=\(\sqrt{x^2-x+1}\)=\(\sqrt{x^2-2.\frac{1}{2}.x+\frac{1}{4}+\frac{3}{4}}\)=\(\sqrt{\left(x-\frac{1}{2}\right)^2+\frac{3}{4}}\) \(\ge\sqrt{\frac{3}{4}}=\frac{\sqrt{3}}{2}\)

Dấu "=" xảy ra <=> x=\(\frac{1}{2}\)

Vậy minG=\(\frac{\sqrt{3}}{2}\) <=> x\(=\frac{1}{2}\)

Câu 1:

Áp dụng BĐT Cô-si:

\(A=\sqrt{\left(2-x\right)\left(2+x\right)}\le\frac{2-x+2+x}{2}=2\)

Dấu "=" xảy ra \(\Leftrightarrow2-x=2+x\Leftrightarrow x=0\)

Câu 2:

\(B=\sqrt{-x^2+x+\frac{1}{4}}\)

\(B=\sqrt{-\left(x^2-x-\frac{1}{4}\right)}\)

\(B=\sqrt{-\left(x^2-x+\frac{1}{4}-\frac{1}{2}\right)}\)

\(B=\sqrt{-\left[\left(x-\frac{1}{2}\right)^2-\frac{1}{2}\right]}\)

\(B=\sqrt{\frac{1}{2}-\left(x-\frac{1}{2}\right)^2}\le\sqrt{\frac{1}{2}}=\frac{\sqrt{2}}{2}\)

Dấu "=" xảy ra \(\Leftrightarrow x=\frac{1}{2}\)

I:
a: \(=x^2-2x+1+x^2-4x+4\)

\(=2x^2-6x+5\)

\(=2\left(x^2-3x+\dfrac{5}{2}\right)\)

\(=2\left(x^2-3x+\dfrac{9}{4}+\dfrac{1}{4}\right)\)

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

Dấu = xảy ra khi x=3/2

b: \(=-4\left(x^2-2x+\dfrac{3}{4}\right)\)

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

Dấu = xảy ra khi x=1

\(P=\dfrac{x-1+4}{\sqrt{x}+1}=\sqrt{x}-1+\dfrac{4}{\sqrt{x}+1}\)

\(=\sqrt{x}+1+\dfrac{4}{\sqrt{x}+1}-2>=2\cdot2-2=2\)

Dấu = xảy ra khi x=1

\(P=A:B=\dfrac{\sqrt{x}+2}{\sqrt{x}}:\dfrac{\sqrt{x}+2}{\sqrt{x}+1}=\dfrac{\sqrt{x}+1}{\sqrt{x}}\)

P>3/2

=>P-3/2>0

=>\(\dfrac{\sqrt{x}+1}{\sqrt{x}}-\dfrac{3}{2}>0\)

=>\(\dfrac{2\sqrt{x}+2-3\sqrt{x}}{2\sqrt{x}}>0\)

=>-căn x+2>0

=>-căn x>-2

=>0<x<4

\(a,\dfrac{\sqrt{a}}{\sqrt{a}-3}-\dfrac{3}{\sqrt{a}+3}-\dfrac{a-2}{a-9}\left(dkxd:a\ne9,a\ge0\right)\)

\(=\dfrac{\sqrt{a}}{\sqrt{a}-3}-\dfrac{3}{\sqrt{a}+3}-\dfrac{a-2}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}\)

\(=\dfrac{\sqrt{a}\left(\sqrt{a}+3\right)-3\left(\sqrt{a}-3\right)-a+2}{a-9}\)

\(=\dfrac{a+3\sqrt{a}-3\sqrt{a}+9-a+2}{a-9}\)

\(=\dfrac{11}{a-9}\)

\(b,\dfrac{x+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}+1}{x+\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\left(dkxd:x\ge0,x\ne1\right)\)

\(=\dfrac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\sqrt{x}+1}{x+\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\)

\(=\dfrac{x+2+\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)-\left(x+\sqrt{x}+1\right)}{x\sqrt{x}-1}\)

\(=\dfrac{x+2+x-1-x-\sqrt{x}-1}{x\sqrt{x}-1}\)

\(=\dfrac{x-\sqrt{x}}{x\sqrt{x}-1}\)

\(=\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(x+\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}\\ =\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\)

\(\dfrac{\sqrt{a}}{\sqrt{a}-3}-\dfrac{3}{\sqrt{a}+3}-\dfrac{a-2}{a-9}\left(\text{đ}k\text{x}\text{đ}:a\ge0;a\ne9\right)\\ =\dfrac{\sqrt{a}\left(\sqrt{a}+3\right)}{\left(\sqrt{a-3}\right)\left(\sqrt{a+3}\right)}-\dfrac{3\left(\sqrt{a}-3\right)}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-3\right)}-\dfrac{a-2}{\left(\sqrt{a}+3\right)\left(\sqrt{a-3}\right)}\\ =\dfrac{a+3\sqrt{a}-\left(3\sqrt{a}-9\right)-\left(a-2\right)}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}\\ =\dfrac{a+3\sqrt{a}-3\sqrt{a}+9-a+2}{\left(\sqrt{a}-3\right)\left(\sqrt{a}+3\right)}\\ =\dfrac{11}{\left(\sqrt{a}-3\right)\left(\sqrt{a+3}\right)}\)

\(b,\dfrac{a+2}{x\sqrt{x}-1}+\dfrac{\sqrt{x}+1}{x\sqrt{x}+1}-\dfrac{1}{\sqrt{x}-1}\left(\text{đ}k\text{x}\text{đ}:x\ge0;x\ne1\right)\\ =\dfrac{x+2}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}+\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}-\dfrac{1\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\\ =\dfrac{x+2+\left(\sqrt{x}+1\right)\left(\sqrt{x-1}\right)-\left(x+\sqrt{x}+1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x+1}\right)}\\ =\dfrac{x+2+x-1-x-\sqrt{x}-1}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\)

\(=\dfrac{x-\sqrt{x}}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\\ =\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}\\ =\dfrac{\sqrt{x}}{x+\sqrt{x}+1}\)