x - √x ( x>=0 )

= ( x - √x + 1/4 ) - 1/4

= ( √x - 1/2 )^2 - 1/4 >= -1/4

Dấu = xảy ra khi : √x - 1/2 = 0

Hay x = 1/4 ( nhận )

Vậy min = -1/4 tại x = 1/4

 Ta có \(\dfrac{a^3+b^3}{2ab}\ge\dfrac{ab\left(a+b\right)}{2ab}=\dfrac{a+b}{2}\) 

(áp dụng BĐT quen thuộc \(a^3+b^3\ge ab\left(a+b\right)\))

 Lập 2 BĐT tương tự rồi cộng theo vế:

 \(VT\ge\dfrac{a+b}{2}+\dfrac{b+c}{2}+\dfrac{c+a}{2}=a+b+c\)

 Dấu "=" xảy ra khi \(a=b=c\)

 Ta có đpcm.

 Dựng hình bình hành ABPC. Khi đó \(AD=AB+CD=CP+CD=DP\)

 Ta có \(\dfrac{AB}{FE}=\dfrac{DA}{DF}\), \(\dfrac{CD}{FE}=\dfrac{DA}{AF}\)

 \(\Rightarrow\dfrac{AB+CD}{FE}=DA\left(\dfrac{1}{DF}+\dfrac{1}{AF}\right)\)

\(\Rightarrow\dfrac{1}{FE}=\dfrac{DA}{DF.AF}\) \(\Rightarrow\dfrac{DF}{FE}=\dfrac{DP}{FA}\) \(\Rightarrow\dfrac{DF}{DC}=\dfrac{DP}{DA}=1\)

 Từ đó \(\Delta DFC\) cân tại D. \(\Rightarrow\widehat{DFC}=\widehat{DCF}=\widehat{CFE}\) \(\Rightarrow\) FC là tia phân giác của \(\widehat{DFE}\). CMTT, FB là tia phân giác của \(\widehat{AFE}\). Do đó \(\widehat{BFC}=90^o\) (đpcm)

Ảnh minh họa:

Với AC là đoạn máy bay cần bay và BC là độ mà máy bay đạt được

Ta có: \(sinA=\dfrac{BC}{AC}\Rightarrow AC=\dfrac{2000}{sin25^o}\approx4732,4\left(m\right)\)

Vậy để đạt độ cao 2000m thì máy bay cần bay khoảng 4732,4m 

a) \(\sqrt{4x^2-4x+1}=5\)

\(\Leftrightarrow\sqrt{\left(2x-1\right)^2}=5\)

\(\Leftrightarrow\left|2x-1\right|=5\)

\(\Leftrightarrow\left[{}\begin{matrix}2x-1=5\left(x\ge\dfrac{1}{2}\right)\\2x-1=-5\left(x< \dfrac{1}{2}\right)\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}2x=6\\2x=-4\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=3\left(tm\right)\\x=-2\left(tm\right)\end{matrix}\right.\)

 b) \(3\sqrt{x-2}-\sqrt{4x-8}+4\sqrt{\dfrac{9x-18}{4}}=14\) \(\left(x\ge2\right)\)

\(\Leftrightarrow3\sqrt{x-2}-\sqrt{4\left(x-2\right)}+4\cdot\dfrac{\sqrt{9x-18}}{2}=14\)

\(\Leftrightarrow3\sqrt{x-2}-2\sqrt{x-2}+2\sqrt{9\left(x-2\right)}=14\)

\(\Leftrightarrow\sqrt{x-2}+6\sqrt{x-2}=14\)

\(\Leftrightarrow7\sqrt{x-2}=14\)

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

\(\Leftrightarrow x-2=4\)

\(\Leftrightarrow x=6\left(tm\right)\)

c) \(\sqrt[3]{4x-1}=3\)

\(\Leftrightarrow4x-1=3^3\)

\(\Leftrightarrow4x-1=27\)

\(\Leftrightarrow4x=27+1\)

\(\Leftrightarrow4x=28\)

\(\Leftrightarrow x=7\)

\(a.\sqrt{4x^2-4x+1}=5\\ \Leftrightarrow\sqrt{\left(2x-1\right)^2}=5\left(ĐK:\left(2x-1\right)^2\ge0\forall x\right)\\ \Leftrightarrow\left|2x-1\right|=5\\ \Leftrightarrow\left[{}\begin{matrix}2x-1=5\\2x-1=-5\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}2x=5+1\\2x=-5+1\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}2x=6\\2x=-4\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}x=3\\x=-2\end{matrix}\right.\\ Vậy.S=\left\{3;-2\right\}\\ b.3\sqrt{x-2}-\sqrt{4x-8}+4\sqrt{\dfrac{9x-18}{4}}=14\\ \Leftrightarrow3\sqrt{x-2}-\sqrt{4\left(x-2\right)}+\dfrac{4\sqrt{9\left(x-2\right)}}{\sqrt{4}}=14\\ \Leftrightarrow3\sqrt{x-2}-2\sqrt{x-2}+6\sqrt{x-2}=14\\ \Leftrightarrow7\sqrt{x-2}=14\left(ĐK:x\ge2\right)\\ \Leftrightarrow\sqrt{x-2}=2\\ \Leftrightarrow x-2=4\\ \Leftrightarrow x=4+2\\ \Leftrightarrow x=6\left(tm\right)\\ Vậy,S=\left\{6\right\}\)

\(c.\sqrt[3]{4x-1}=3\\ \Leftrightarrow4x-1=27\\ \Leftrightarrow4x=27+1\\ \Leftrightarrow4x=28\\ \Leftrightarrow x=7\)

Ta có:

\(a^3+b^3=2\)

\(\Rightarrow\left(a+b\right)\left(a^2-ab+b^2\right)=2\)

\(\Rightarrow a+b=\dfrac{2}{a^2-ab+b^2}\)

Mà: \(2\left(a-b\right)^2\ge0\forall a,b\)

\(\Rightarrow2\left(a^2-2ab+b^2\right)\ge0\)

\(\Rightarrow2a^2-4ab+2b^2\ge0\)

\(\Rightarrow2a^2+2a^2-4ab+2b^2+2b^2\ge2a^2+2b^2\)

\(\Rightarrow4a^2-4ab+4b^2\ge2\left(a+b\right)^2\)

\(\Rightarrow4\left(a^2-ab+b^2\right)\ge2\left(a+b\right)^2\ge\left(a+b\right)^2\)

\(\Rightarrow a^2-ab+b^2\ge\dfrac{\left(a+b\right)^2}{4}\)

\(\Rightarrow\dfrac{2}{a^2-ab+b^2}\le\dfrac{8}{\left(a+b\right)^2}\)

\(\Rightarrow a+b\le\dfrac{8}{\left(a+b\right)^2}\)

\(\Rightarrow\left(a+b\right)^3\le8\)

\(\Rightarrow a+b\le2\)

Vậy: \(A_{max}=2\)

Đặt \(f\left(x\right)=ax^3+bx^2+cx+d\left(a\inℤ^+\right)\)

\(f\left(5\right)=125a+25b+5c+d\)

\(f\left(3\right)=27a+9b+3c+d\)

\(\Rightarrow f\left(5\right)-f\left(3\right)=98a+16b+2c\)

Mà \(f\left(5\right)-f\left(3\right)=2022\) nên \(98a+16b+2c=2022\) 

\(\Leftrightarrow49a+8b+c=1011\)

Lại có \(f\left(7\right)=343a+49b+7c+d\)

\(f\left(1\right)=a+b+c+d\)

\(\Rightarrow f\left(7\right)-f\left(1\right)=342a+48b+6c\) \(=6\left(57a+8b+c\right)\) \(=6\left(8a+1011\right)\) (vì \(49a+8b+c=1011\))

 Mà do \(a\inℤ^+\) nên \(f\left(7\right)-f\left(1\right)\) là hợp số (đpcm)

công thức tổng quát: f(x)=x^{3        sdasdasdadasd}

\(VT\ge\dfrac{4}{x^2+y^2+2xy}=\dfrac{4}{\left(x+y\right)^2}\ge4\) (vì \(x+y\le1\) )

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

Ta có đpcm

\(A=\dfrac{1}{x^2+y^2}+\dfrac{1}{2xy}+\dfrac{1}{4xy}+4xy+\dfrac{5}{4xy}\)

\(\ge\dfrac{4}{x^2+y^2+2xy}+2\sqrt{\dfrac{1}{4xy}.4xy}+\dfrac{5}{4.\dfrac{\left(x+y\right)^2}{4}}\)

\(\ge\dfrac{4}{1^2}+2+\dfrac{5}{1^2}\) (do \(x+y\le1\))

\(=11\)

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

Vậy GTNN của A là 11.