\(=\sqrt{\left(5x-2\right)^2}+\sqrt{\left(5x\right)^2}\)= \(\left|2-5x\right|+\left|5x\right|\ge2+5x-5x=2\)

min A=2 \(\Leftrightarrow\hept{\begin{cases}2-5x\ge0\\5x\ge0\end{cases}\Leftrightarrow0\le x\le\frac{2}{5}}\)

Chuẩn đấy

Ta có : M =\(\sqrt{\left(5x\right)^2-2.2.5x+2^2}+\sqrt{\left(5x\right)^2}\)= \(\sqrt{\left(5x-2\right)^2}+\left|5x\right|\) = \(\left|5x-2\right|+\left|5x\right|=\left|2-5x\right|+\left|5x\right|\ge\left|2-5x+5x\right|=\left|2\right|=2\)

=> M ≥ 2

Min M = 2 : dấu"="xảy ra khi: ....

Mình bận rồi tự làm tí nhé!!

Lời giải:

Ta có:

\(C=\sqrt{(5x-2)^2}+\sqrt{(5x)^2}\)

\(=|5x-2|+|5x|=|2-5x|+|5x|\)

Áp dụng BĐT \(|a|+|b|\geq |a+b|\) ta có:

\(C=|2-5x|+|5x|\geq |2-5x+5x|=2\)

Vậy \(C_{\min}=2\). Dấu "=" xảy ra khi \(5x(2-5x)\geq 0\Leftrightarrow 0\leq x\leq \frac{2}{5}\)

\(B=\left|5x-2\right|+\left|5x-3\right|\)

\(=\left|5x-2\right|+\left|3-5x\right|\)

=>B>=|5x-2+3-5x|=1

Dấu = xảy ra khi (5x-2)(5x-3)<=0

=>2/5<=x<=3/5

\(A=5x^2-25x+35+7y^8\)

\(=5\left(x^2-5x+7\right)+7y^8\)

\(=5\left(x^2-5x+\frac{25}{4}+\frac{3}{4}\right)+7y^8\)

\(=5\left[\left(x-\frac{5}{2}\right)^2+\frac{3}{4}\right]+7y^8\)

\(=5\left(x-\frac{5}{2}\right)^2+\frac{15}{4}+7y^8\ge\frac{15}{4}\)

\(\Leftrightarrow x=\frac{5}{2};y=0\)

\(A=9x^2+18x-20\)

\(\Leftrightarrow A=\left(3x\right)^2+2.3x.3+9-29\)

\(\Leftrightarrow A=\left(3x+3\right)^2-29\le-29\forall x\)

Dấu " = " xảy ra

\(\Leftrightarrow\left(3x+3\right)^2=0\Leftrightarrow3x+3=0\Leftrightarrow3x=-3\Leftrightarrow x=-1\)

Vậy Min A là : \(-29\Leftrightarrow x=-1\)

\(B=m^2+10m+1\)

\(\Leftrightarrow B=m^2+2.m.5+25-24\)

\(\Leftrightarrow B=\left(m+5\right)^2-24\le-24\forall m\)

Dấu \("="\) xảy ra

\(\Leftrightarrow\left(m+5\right)^2=0\Leftrightarrow m+5=0\Leftrightarrow m=-5\)

Vậy Min B là : -24 \(\Leftrightarrow m=-5\)

\(C=25x^2-20x+30\)

\(\Leftrightarrow C=\left(5x\right)^2-2.5x.2+4+26\)

\(\Leftrightarrow C=\left(5x-2\right)^2+26\le26\forall x\)

Dấu " = " xảy ra

\(\Leftrightarrow\left(5x-2\right)^2=0\Leftrightarrow5x-2=0\Leftrightarrow5x=2\Leftrightarrow x=\dfrac{2}{5}\)

Vậy Min C là : 26 \(\Leftrightarrow x=\dfrac{2}{5}\)

Sorry , sửa lại :

\(\le\) \(\Rightarrow\) \(\ge\)

\(M=\sqrt{x^2+y^2-2xy+2x-2y+10}+2y^2-8y+2024\\ =\sqrt{\left(x^2+y^2+1-2xy+2x-2y\right)+9}+\left(2y^2-8y+8\right)+2016\\ =\sqrt{\left(x-y+1\right)^2+9}+2\left(y^2-4y+4\right)+2016\\ =\sqrt{\left(x-y+1\right)^2+9}+2\left(y-2\right)^2+2016\) \(\text{Do }\left(x-y+1\right)^2\ge0\forall x;y\\ \Rightarrow\left(x-y+1\right)^2+9\ge9\forall x;y\\ \Rightarrow\sqrt{\left(x-y+1\right)^2+9}\ge3\forall x;y\\ Mà\text{ }2\left(y-2\right)^2\ge0\forall y\\ \Rightarrow\sqrt{\left(x-y+1\right)^2+9}+2\left(y-2\right)^2\ge3\forall x;y\\ M=\sqrt{\left(x-y+1\right)^2+9}+2\left(y-2\right)^2+2016\ge2019\forall x;y\)

Dấu "=" xảy ra khi:

\(\left\{{}\begin{matrix}2\left(y-2\right)^2=0\\\left(x-y+1\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y-2=0\\x-y+1=0\end{matrix}\right.\\ \Leftrightarrow\left\{{}\begin{matrix}y=2\\x=y-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=2\\x=1\end{matrix}\right.\)

Vậy \(M_{Min}=2019\) khi \(\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)

\(Q=\sqrt{25x^2-20x+4}+\sqrt{25x^2-30x+9}\\ =\sqrt{\left(5x-2\right)^2}+\sqrt{\left(5x-3\right)^2}\\ =\left|5x-2\right|+\left|5x-3\right|\\ =\left|5x-2\right|+\left|3-5x\right|\)

Áp dụng BDT: \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\)

\(\Rightarrow\left|5x-2\right|+\left|3-5x\right|\ge\left|5x-2+3-5x\right|=\left|1\right|=1\)

Dấu "=" xảy ra khi:

\(\left(5x-2\right)\left(3-5x\right)\ge0\\\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}5x-2\ge0\\3-5x\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}5x-2\le0\\3-5x\le0\end{matrix}\right.\end{matrix}\right. \) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}5x\ge2\\5x\le3\end{matrix}\right.\\\left\{{}\begin{matrix}5x\le2\\5x\ge3\end{matrix}\right.\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge\dfrac{2}{5}\\x\le\dfrac{3}{5}\end{matrix}\right.\left(T/m\right)\\\left\{{}\begin{matrix}x\le\dfrac{2}{5}\\x\ge\dfrac{3}{5}\end{matrix}\right.\left(K^0\text{ }T/m\right)\end{matrix}\right.\)

\(\Leftrightarrow\dfrac{2}{5}\le x\le\dfrac{3}{5}\)

Vậy \(Q_{Min}=1\) khi \(\dfrac{2}{5}\le x\le\dfrac{3}{5}\)

Ta có: \(x=-24\Leftrightarrow-x=24\Leftrightarrow1-x=25\)

Thay vào E ta được:

\(E=x^{20}+\left(1-x\right)x^{19}+\left(1-x\right)x^{18}+...+\left(1-x\right)x^2+\left(1-x\right)x+\left(1-x\right)\)

\(E=x^{20}+x^{19}-x^{20}+x^{18}-x^{19}+...+x^2-x^3+x-x^2+1-x\)

\(E=1\)

câu a) rút x theo y thế vào A rồi áp dụng HĐT

b)rút xy thế vào B 

c)HĐT

d)rút x theo y thé vào C

rồi dùng BĐT cô-si

e)BĐT chưa dấu giá trị tuyệt đối

\(C=\sqrt{25x^2-20x+4}+\sqrt{25x^2}\)

\(C=\sqrt{\left(5x-2\right)^2}+\sqrt{\left(5x\right)^2}\)

\(C=\left|5x-2\right|+\left|5x\right|=\left|2-5x\right|+\left|5x\right|\)

\(C\ge\left|2-5x+5x\right|=2\)

Dấu " = " xảy ra \(\Leftrightarrow\)( 2 - 5x ) . 5x \(\ge\)0

\(\Leftrightarrow\)\(\hept{\begin{cases}x\ge0\\2-5x\ge0\end{cases}}\)hoặc \(\hept{\begin{cases}x\le0\\2-5x\le0\end{cases}}\)

\(\Leftrightarrow\)\(0\le x\le\frac{2}{5}\)

Vậy GTNN của C là 2 \(\Leftrightarrow\)\(0\le x\le\frac{2}{5}\)

\(C=\sqrt{25x^2-20x+4}+\sqrt{25x^2}\)

\(C=\sqrt{\left(5x-2\right)^2}+\sqrt{\left(5x\right)^2}\)

\(C=\left|5x-2\right|+\left|5x\right|\)

\(C=\left|2-5x\right|+\left|5x\right|\ge\left|2-5x+5x\right|=2\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}2-5x\ge0\\5x\ge0\end{cases}\Leftrightarrow\hept{\begin{cases}x\le\frac{2}{5}\\x\ge0\end{cases}\Leftrightarrow0\le}x\le\frac{2}{5}}\)