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\(J=\frac{2010}{4x+20\sqrt{x}+30}\)
\(=\frac{2010}{\left(2\sqrt{x}\right)^2+2.2\sqrt{x}.5+25+5}\)
\(=\frac{2010}{\left(2\sqrt{x}+5\right)^2+5}\)
\(A_{max}\Leftrightarrow\frac{2010}{\left(2\sqrt{x}+5\right)^2+5}\)lớn nhất
\(\Rightarrow\left(2\sqrt{x}+5\right)^2+5\)nhỏ nhất
\(\Rightarrow\left(2\sqrt{x}+5\right)^2\)nhỏ nhất
Mà \(2\sqrt{x}+5\ge5\Rightarrow2\sqrt{x}+5=5\Leftrightarrow2\sqrt{x}=0\Leftrightarrow x=0\)
Với x = 0 \(\Rightarrow J_{max}=\frac{2010}{4.0+20\sqrt{0}+30}=\frac{2010}{30}=67\)
k) ĐK: $x^2\geq 5$
PT $\Leftrightarrow 2\sqrt{x^2-5}-\frac{1}{3}\sqrt{x^2-5}+\frac{3}{4}\sqrt{x^2-5}-\frac{5}{12}\sqrt{x^2-5}=4$
$\Leftrightarrow 2\sqrt{x^2-5}=4$
$\Leftrightarrow \sqrt{x^2-5}=2$
$\Rightarrow x^2-5=4$
$\Leftrightarrow x^2=9\Rightarrow x=\pm 3$ (đều thỏa mãn)
l) ĐKXĐ: $x\geq -1$
PT $\Leftrightarrow 2\sqrt{x+1}+3\sqrt{x+1}-\sqrt{x+1}=4$
$\Leftrightarrow 4\sqrt{x+1}=4$
$\Leftrightarrow \sqrt{x+1}=1$
$\Rightarrow x+1=1$
$\Rightarrow x=0$
m)
ĐKXĐ: $x\geq -1$
PT $\Leftrightarrow 4\sqrt{x+1}+2\sqrt{x+1}=16-\sqrt{x+1}+3\sqrt{x+1}$
$\Leftrightarrow 6\sqrt{x+1}=16+2\sqrt{x+1}$
$\Leftrightarrow 4\sqrt{x+1}=16$
$\Leftrightarrow \sqrt{x+1}=4$
$\Rightarrow x=15$ (thỏa mãn)
h)
ĐKXĐ: $x\geq -5$
PT $\Leftrightarrow \sqrt{x+5}=6$
$\Rightarrow x+5=36\Rightarrow x=31$ (thỏa mãn)
i) ĐKXĐ: $x\geq 5$
PT \(\Leftrightarrow \sqrt{x-5}+4\sqrt{x-5}-\sqrt{x-5}=12\)
\(\Leftrightarrow 4\sqrt{x-5}=12\Leftrightarrow \sqrt{x-5}=3\Rightarrow x-5=9\Rightarrow x=14\) (thỏa mãn)
j)
ĐKXĐ: $x\geq 0$
PT $\Leftrightarrow 3\sqrt{2x}+\sqrt{2x}-6\sqrt{2x}+4=0$
$\Leftrightarrow -2\sqrt{2x}+4=0$
$\Leftrightarrow \sqrt{2x}=2$
$\Rightarrow x=2$ (thỏa mãn)
Có: \(C=\frac{1}{\sqrt{x^2-4x+5}}\)
\(\Leftrightarrow C=\frac{1}{\sqrt{\left(x-2\right)^2+1}}\)\(\le1\)
Vậy Cmin=1 \(\Leftrightarrow x=2\)
Có: \(B=5-\sqrt{x^2-6x+14}\)
\(\Leftrightarrow B=5-\sqrt{\left(x-3\right)^2+5}\) \(\le5-\sqrt{5}\)
Vậy \(B_{min}=5-\sqrt{5}\Leftrightarrow x=3\)
a) Ta có: \(x-\sqrt{x}+1=\left(\sqrt{x}-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\left(\forall x\right)\)
=> \(A=\frac{1}{x-\sqrt{x}+1}\le\frac{1}{\frac{3}{4}}=\frac{4}{3}\left(\forall x\right)\)
Dấu "=" xảy ra khi: \(\left(\sqrt{x}-\frac{1}{2}\right)^2=0\Rightarrow x=\frac{1}{4}\)
Vậy Max(A) = 4/3 khi x = 1/4
b) \(B=\sqrt{4x-x^2+21}=\sqrt{-\left(x^2-4x+4\right)+25}\)
\(=\sqrt{25-\left(x-2\right)^2}\le\sqrt{25}=5\left(\forall x\right)\)
Dấu "=" xảy ra khi: \(\left(x-2\right)^2=0\Rightarrow x=2\)
Vậy Max(B) = 5 khi x = 2
c) \(C=1+\sqrt{-9x^2+6x}=1+\sqrt{-\left(9x^2-6x+1\right)+1}\)
\(=1+\sqrt{1-\left(3x-1\right)^2}\le1+\sqrt{1}=2\)
Dấu "=" xảy ra khi: \(\left(3x-1\right)=0\Rightarrow x=\frac{1}{3}\)
Vậy Max(C) = 2 khi x = 1/3
d) Ta có: \(D=\sqrt{x-2}+\sqrt{4-x}\)
=> \(D^2=\left(\sqrt{x-2}+\sqrt{4-x}\right)\le\left(1^2+1^2\right)\left(x-2+4-x\right)\) ( BĐT Bunhia)
\(=2.2=4\)
=> \(D\le2\left(\forall x\right)\)
Dấu "=" xảy ra khi: \(x-2=4-x\Rightarrow x=3\)
Vậy Max(D) = 2 khi x = 3
