\(y=\sqrt{\left(\frac{x}{2}\right)^2-2\cdot\frac{x}{2}\cdot\frac{1}{6}+\frac{1}{36}+\frac{35}{36}}=\sqrt{\left(\frac{x}{2}-\frac{1}{6}\right)^2+\frac{35}{36}}\ge\sqrt{\frac{35}{36}}=\frac{\sqrt{35}}{6}\)
Dấu \("="\Leftrightarrow x=\frac{1}{3}\)
\(y=\sqrt{\left(\frac{x}{2}\right)^2-2\cdot\frac{x}{2}\cdot\frac{1}{6}+\frac{1}{36}+\frac{35}{36}}=\sqrt{\left(\frac{x}{2}-\frac{1}{6}\right)^2+\frac{35}{36}}\ge\sqrt{\frac{35}{36}}=\frac{\sqrt{35}}{6}\)
Dấu \("="\Leftrightarrow x=\frac{1}{3}\)
Tìm max hoặc min của biểu thức sau:
\(C=\sqrt{2x^2+y^2-4x+2y+3}+\sqrt{3x^2+y^2-6x-8y+19}\)
\(D=\frac{1}{x}\sqrt{\frac{x-1}{x^2-4x+29}}+\frac{1}{y}\sqrt{\frac{y-25}{y^2-100y+2501}}\)
Tìm min:
a, \(A=x+\frac{x-1}{\sqrt{x^2-2x}}\) với x>2
b, \(B=x\sqrt{x}-6x+13\sqrt{x}+\frac{4}{\sqrt{x}}\)
c,\(C=\frac{1-4\sqrt{x}}{2x+1}-\frac{2x}{x^2+1}\)
Bài 1 : Giải các phương trình sau: x\(\sqrt{x^2+16}+x^2\)=24 ; \(\sqrt{x-y}=3-\left(\sqrt{x-y-9}\right)^2\)
Bài 2 : Tìm Min A, biết A= \(\frac{a^2}{b-1}+\frac{b^2}{a-1}\left(a,b>1\right)\)
Bài 3: Cho M = \( \left( \sqrt{x+4\sqrt{x-4}}+\sqrt{x-4\sqrt{x-4}}\right)\): \(\left|\frac{4-x}{x}\right|\)
1 . Cho x+y+z=xyz. Tìm Min A= \(\frac{y}{x\sqrt{y^2+1}}+\frac{z}{y\sqrt{z^2+1}}+\frac{x}{z\sqrt{x^2+1}}\)
2 . Cho a,b,c>0 thỏa a+b+c=3, tìm GTNN
\(P=\frac{25a^2}{\sqrt{2a^2+16ab+7b^2}}+\frac{25b^2}{\sqrt{2b^2+16bc+7c^2}}+\frac{c^2\left(a+3\right)}{a}\)
Cho 3 số thực dương x,z,y tm x+y+z=\(\sqrt{2}\). Tìm MIN T=\(\sqrt{(x+y)(y+z)(x+z)}(\frac{\sqrt{y+z}}{x}+\frac{\sqrt{y+x}}{z}+\frac{\sqrt{x+z}}{y})\)
Rút gọn:
a,\(\frac{3+\sqrt{3}}{1+\sqrt{3}}\)
b,\(\frac{2\sqrt{3}-\sqrt{6}}{\sqrt{8}-\sqrt{2}}\)
c,\(\frac{y-2\sqrt{y}}{\sqrt{y}-2}\)
d,\(\frac{x+2\sqrt{x}-3}{\sqrt{x}-1}\)
e,\(\frac{4y+3\sqrt{y}-7}{4\sqrt{y}+7}\)
g,\(\frac{x-3\sqrt{x}-4}{x-\sqrt{x}-12}\)
1.Giải hpt
\(\left\{{}\begin{matrix}\frac{7}{\sqrt{x+7}}-\frac{4}{\sqrt{y-6}}=\frac{-1}{4}\\\frac{5}{\sqrt{x+7}}+\frac{3}{\sqrt{y-6}}=\frac{11}{4}\end{matrix}\right.\)
Giải phương trình:
a) \(2\sqrt{x^2-4}-3=6\sqrt{x-2}-\sqrt{x+2}\)
b) \(\frac{\sqrt{x-2016}-1}{x-2016}+\frac{\sqrt{y-2017}-1}{y-2017}+\frac{\sqrt{z-2018}-1}{z-2018}=\frac{3}{4}\)
c) \(\sqrt{3+\sqrt{3+x}}=x\)
d) \(\sqrt{6x^2+1}=\sqrt{2x-3}+x^2\)
e) \(\sqrt{x^2+3x+5}+\sqrt{x^2-2x+5}=5\sqrt{x}\)
f) \(\sqrt{x^2+3x}+2\sqrt{x+2}=2x+\sqrt{x+\frac{6}{x}+5}\)
Rút gọn biểu thức sau :( chú ý đặt ĐKXĐ trước khi trước khi thực hiện rút gọn)
a,P= \(\frac{3\sqrt{x}+2}{2\sqrt{x}-1}+\frac{\sqrt{x}-1}{\sqrt{x}+4}-\frac{x-6\sqrt{x}+5}{2x+7\sqrt{x}-4}\)
b, D=\(\frac{\sqrt{x}+4}{1-7\sqrt{x}}+\frac{\sqrt{x}-2}{\sqrt{x+1}}+\frac{24\sqrt{x}}{7x+6\sqrt{x}-1}\)