Giups mik vs

b. + Vì \(|6-2x|\ge0\)\(\forall x\)

\(\Rightarrow\)\(|6-2x|-5\ge0-5\)\(\forall x\)

\(\Rightarrow\)B\(\ge\)-5 \(\forall x\)

Vậy GTNN của B= -5 \(\Leftrightarrow\)6-2x=0

                                    \(\Leftrightarrow\)2x=6

                                   \(\Leftrightarrow\)x=3

+ Vì -\(|6-2x|\le0\forall x\)

\(\Rightarrow\)\(|6-2x|-5\le0+5\forall x\)

\(\Rightarrow B\le5\forall x\)

Vậy GTLN của B= 5 \(\Leftrightarrow6-2x=0\)

                                \(\Leftrightarrow2x=1\)

                                \(\Leftrightarrow x=\frac{1}{2}\)

c,+ Vì \(|x+1|\ge0\forall x\)

\(\Rightarrow\)\(3-|x+1|\ge3-0\forall x\)

\(\Rightarrow C\ge3\forall x\)

Vậy GTNN của C=3 \(\Leftrightarrow x+1=0\)

                                 \(\Leftrightarrow x=-1\)

+ Vì \(-|x+1|\le0\forall x\)

\(\Rightarrow3-|x+1|\le3+0\forall x\)

\(\Rightarrow C\le3\forall x\)

Vậy GTLN của \(C=3\Leftrightarrow x+1=0\)

                                     \(\Leftrightarrow x=-1\)

Mình chỉ làm vậy thôi nhé!

THANKS  BẠN NHIỀU NHA

không có j

Ta sẽ chứng minh \(P_{min}=1\)

TH1: \(xyz=0\)

\(\Rightarrow x^2y^2z^2=0\Rightarrow x^4+y^4+z^4=1\)

\(P=x^2+y^2+z^2\ge\sqrt{x^4+y^4+z^4}=1\)

TH2: \(xyz\ne0\) , từ điều kiện, tồn tại 1 tam giác nhọn ABC sao cho \(\left\{{}\begin{matrix}x^2=cosA\\y^2=cosB\\z^2=cosC\end{matrix}\right.\)

\(P=cosA+cosB+cosC-\sqrt{2cosA.cosB.cosC}\)

Ta sẽ chứng minh \(cosA+cosB+cosC-\sqrt{2cosA.cosB.cosC}\ge1\)

\(\Leftrightarrow4sin\dfrac{A}{2}sin\dfrac{B}{2}sin\dfrac{C}{2}\ge\sqrt{2cosA.cosB.cosC}\)

\(\Leftrightarrow8sin^2\dfrac{A}{2}sin^2\dfrac{B}{2}sin^2\dfrac{C}{2}\ge cosA.cosB.cosC\)

\(\Leftrightarrow\dfrac{8sin^2\dfrac{A}{2}sin^2\dfrac{B}{2}sin^2\dfrac{C}{2}}{8sin\dfrac{A}{2}sin\dfrac{B}{2}sin\dfrac{C}{2}cos\dfrac{A}{2}cos\dfrac{B}{2}cos\dfrac{C}{2}}\ge cotA.cotB.cotC\)

\(\Leftrightarrow tan\dfrac{A}{2}tan\dfrac{B}{2}tan\dfrac{C}{2}\ge cotA.cotB.cotC\)

\(\Leftrightarrow tanA.tanB.tanC\ge cot\dfrac{A}{2}cot\dfrac{B}{2}cot\dfrac{C}{2}\)

\(\Leftrightarrow tanA+tanB+tanC\ge cot\dfrac{A}{2}+cot\dfrac{B}{2}+cot\dfrac{C}{2}\)

Ta có:

\(tanA+tanB=\dfrac{sin\left(A+B\right)}{cosA.cosB}=\dfrac{2sinC}{cos\left(A-B\right)-cosC}\ge\dfrac{2sinC}{1-cosC}=\dfrac{2sin\dfrac{C}{2}cos\dfrac{C}{2}}{2sin^2\dfrac{C}{2}}=cot\dfrac{C}{2}\)

Tương tự: \(tanA+tanC\ge cot\dfrac{B}{2}\) ; \(tanB+tanC\ge cot\dfrac{A}{2}\)

Cộng vế với vế ta có đpcm

Vậy \(P_{min}=1\) khi \(\left(x^2;y^2;z^2\right)=\left(1;0;0\right)\) và các hoán vị hoặc \(\left(x^2;y^2;z^2\right)=\left(\dfrac{1}{2};\dfrac{1}{2};\dfrac{1}{2}\right)\)

Bài 1: 

Ta có: \(D=\sqrt{16x^4}-2x^2+1\)

\(=4x^2-2x^2+1\)

\(=2x^2+1\)

a,,A=|x-3|+1

Ta thấy:\(\left|x-3\right|\ge0\)

\(\Rightarrow\left|x-3\right|+1\ge0+1=1\)

\(\Rightarrow A\ge1\).Dấu = khi x=3

Vậy....

b)B=|6-2x|-5

Ta thấy:\(\left|6-2x\right|\ge0\)

\(\Rightarrow\left|6-2x\right|-5\ge0-5=-5\)

\(\Rightarrow B\ge-5\).Dấu = khi x=3

Vậy...

c) C=3-|x+1|

Ta thấy:\(-\left|x+1\right|\le0\)

\(\Rightarrow3-\left|x+1\right|\le3-0=3\)

\(\Rightarrow C\le3\).Dấu = khi x=-1

e) E= -(x+1)^2 -|2-y|+11

Ta thấy:\(\hept{\begin{cases}-\left(x+1\right)^2\\-\left|2-y\right|\end{cases}\le}0\)

\(\Rightarrow-\left(x+1\right)^2-\left|2-y\right|\le0\)

\(\Rightarrow-\left(x+1\right)^2-\left|2-y\right|+11\le0+11=11\)

\(\Rightarrow E\le11\).Dấu = khi x=-1 y=2

Vậy... 

f)F= (x-1)^2+|2y+2|-3

Ta thấy:\(\hept{\begin{cases}\left(x-1\right)^2\\\left|2y+2\right|\end{cases}}\ge0\)

\(\Rightarrow\left(x-1\right)^2+\left|2y+2\right|\ge0\)

\(\Rightarrow\left(x-1\right)^2+\left|2y+2\right|-3\ge0-3=-3\)

\(\Rightarrow F\ge-3\).Dấu = khi x=1  y=-1

Vậy...

Câu 1 :

\(E=4x^2+y^2-4x-2y+3\)

\(E=\left(2x\right)^2-2\cdot2x\cdot1+1^2+y^2-2\cdot y\cdot1+1^2+1\)

\(E=\left(2x-1\right)^2+\left(y-1\right)^2+1\ge1\forall x;y\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}2x-1=0\\y-1=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=\frac{1}{2}\\y=1\end{cases}}\)

Câu 2 :

\(G=x^2+2y^2+2xy-2y\)

\(G=x^2+2xy+y^2+y^2-2.y\cdot1+1^2-1\)

\(G=\left(x+y\right)^2+\left(y-1\right)^2-1\ge-1\forall x;y\)

Dấu "=" xảy ra \(\Leftrightarrow\hept{\begin{cases}x+y=0\\y-1=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+1=0\\y=1\end{cases}\Leftrightarrow}\hept{\begin{cases}x=-1\\y=1\end{cases}}}\)

Còn câu F bạn ơi. Giúp Gk vs

\(F=\frac{3}{2x^2+x+1}=\frac{3}{2\left(x^2+\frac{x}{2}+\frac{1}{2}\right)}=\frac{3}{2\left(x^2+2x\cdot\frac{1}{4}+\frac{1}{16}\right)+\frac{7}{8}}=\frac{3}{2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}}\)

Vi \(2\left(x+\frac{1}{4}\right)^2\ge0\Rightarrow2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}\ge8\)

\(\Rightarrow\frac{1}{2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}}\le\frac{1}{\frac{7}{8}}\Rightarrow F=\frac{3}{2\left(x+\frac{1}{4}\right)^2+\frac{7}{8}}\le\frac{3}{\frac{7}{8}}=\frac{24}{7}\)

Dấu "=" xảy ra <=>x+1/4=0<=>x=-1/4

a) \(A=4x^2-12x+100=\left(2x\right)^2-12x+3^2+91=\left(2x-3\right)^2+91\)

Ta có: \(\left(2x-3\right)^2\ge0\forall x\inℤ\)

\(\Rightarrow\left(2x-3\right)^2+91\ge91\)

hay A \(\ge91\)

Dấu "=" xảy ra <=> \(\left(2x-3\right)^2=0\)

<=> 2x-3=0

<=> 2x=3

<=> \(x=\frac{3}{2}\)

Vậy Min A=91 đạt được khi \(x=\frac{3}{2}\)

b) \(B=-x^2-x+1=-\left(x^2+x-1\right)=-\left(x^2+x+\frac{1}{4}-\frac{5}{4}\right)=-\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\)

Ta có: \(-\left(x+\frac{1}{2}\right)^2\le0\forall x\)

\(\Rightarrow-\left(x+\frac{1}{2}\right)^2+\frac{5}{4}\le\frac{5}{4}\) hay B\(\le\frac{5}{4}\)

Dấu "=" \(\Leftrightarrow-\left(x+\frac{1}{2}\right)^2=0\)

\(\Leftrightarrow x+\frac{1}{2}=0\)

\(\Leftrightarrow x=\frac{-1}{2}\)

Vậy Max B=\(\frac{5}{4}\)đạt được khi \(x=\frac{-1}{2}\)

\(C=2x^2+2xy+y^2-2x+2y+2\)

\(C=x^2+2x\left(y-1\right)+\left(y-1\right)^2+x^2+1\)

\(\Leftrightarrow C=\left(x+y-1\right)^2+x^2+1\)

Ta có: 

\(\hept{\begin{cases}\left(x+y-1\right)^2\ge0\forall x;y\inℤ\\x^2\ge0\forall x\inℤ\end{cases}}\)

\(\Leftrightarrow\left(x+y-1\right)^2+x^2+1\ge1\)

hay C\(\ge\)1

Dấu "=" xảy ra khi \(\hept{\begin{cases}\left(x+y-1\right)^2=0\\x^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+y=1\\x=0\end{cases}\Leftrightarrow}\hept{\begin{cases}y=1\\x=0\end{cases}}}\)

Vậy Min C=1 đạt được khi y=1 và x=0

\(S=\left(x^2+y^2+1+2xy+2x+2y\right)+\left(y^2-4y+4\right)+2021\)

\(S=\left(x+y+1\right)^2+\left(y-2\right)^2+2021\ge2021\)

Dấu "=" xảy ra khi \(\left(x;y\right)=\left(-3;2\right)\)