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/y-3/>=0

A = (x-3)² +/y-3/ +7 -9

GTNN A = -2

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Câu a :

Ta có :

\(x^2-x+3\)

\(=x^2-x+\dfrac{1}{4}+\dfrac{11}{4}\)

\(=\left(x-\dfrac{1}{2}\right)^2+\dfrac{11}{4}\)

Do : \(\left(x-\dfrac{1}{2}\right)^2\ge0\Rightarrow\left(x-\dfrac{1}{2}\right)^2+\dfrac{11}{4}\ge\dfrac{11}{4}\)

Vậy GTNN của biểu thức trên \(=\dfrac{11}{4}\)

Dấu \(=\) xảy ra khi \(\left(x-\dfrac{1}{2}\right)^2=0\Rightarrow x=\dfrac{1}{2}\)

Câu b :

Ta có :

\(-x^2+6-8\)

\(=-x^2+6x-9+1\)

\(=-\left(x^2-6x+9\right)+1\)

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

Do :

\(\left(x-3\right)^2\ge0\Rightarrow-\left(x-3\right)^2\le0\Rightarrow-\left(x-3\right)^2+1\le1\)

Vâỵ GTNN của biểu thức \(=11\)

Dấu \(=\) xảy ra khi \(\left(x-3\right)^2=0\Rightarrow x=3\)

có x^2-x+3=x^2-1/2*2*x+(1/2)^2+11/4=(x-1/2)^2+11/4

có (x-1/2)^2>hoặc =0suy ra(x-1/2)^2+11/4>hoặc=11/4

dấu =xảy ra khi x-1/2=0suy ra x=1/2

có -x^2+6x-8=-(x^2-6x+8)=-(x-3)^2+1

có-(x-3)^2>hoặc=0suy ra -(x-3)^2+1>hoặc=1

dấu = xảy ra khi x-3=0suy ra x=3

=x^2-6x+9+2

=(x-3)^2+2

vậy GTNN là 2

M = x⁴ - 6x³ + 10x² - 6x + 9

M = (x² - 6x + 9) + x⁴ - 6x³ + 9x²

M = (x - 3)² + x²(x² - 6x + 9)

M = (x - 3)².(1 + x²)

Ta có:\(\left(x-3\right)^2\ge0;\left(1+x^2\right)\ge1\)

\(\Rightarrow M\ge1\)

Dấu 'x' xảy ra khi:

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

Vậy Mmin = 1 khi x = 3

Chúc bạn học tốt!!!

Mình giải lại từ dòng số 6 nhé!!!

=> M = 0 

Dấu '=' xảy ra khi:

(x - 3)² = 0 => x - 3 = 0

=> x = 3

Vậy Mmin = 0 khi x = 3

x²+4y²+6x+8y+1

=x²+6x+9+4y²+8y+4-12

=(x+3)²+(2y+2)²-12

\(\Rightarrow\)(x+3)²+(2y+2)²\(\ge\)0 với mọi x,y.

\(\Rightarrow\)(x+3)²+(2y+2)² \(\ge\)-12 với mọi x,y.

Vay GTNN la -12

Dấu "=" xảy ra khi x+3=0 \(\Rightarrow\)x=-3

                           2y+2=0\(\Rightarrow\)y=-1

Nhớ k nha . 

\(A=9x^2+6x-7\)

\(\Rightarrow A=\left(3x\right)^2+2\cdot3x+1-8\)

\(\Rightarrow A=\left(3x+1\right)^2-8\ge-8\)

Vậy GTNN của A là -8

A\(=9x^2+6x-7\)

\(=9\left(x^2+\dfrac{2}{3}x-\dfrac{7}{9}\right)\)

\(=9\left(x^2+2.x.\dfrac{1}{3}+\dfrac{1}{9}+\dfrac{-8}{9}\right)\)

\(=9\left(x+\dfrac{1}{3}\right)^2+\left(-8\right)\)

Vì \(\left(x+\dfrac{1}{3}\right)^2\ge0\)

\(\Rightarrow\left(x+\dfrac{1}{3}\right)^2+\left(-8\right)\ge-8\)

Dấu = xảy ra khi x+\(\dfrac{1}{3}=0\Rightarrow x=\dfrac{-1}{3}\)

Vậy GTNN của A=-8 khi x=\(\dfrac{-1}{3}\)

\(A=4-6x-x^2=-\left(x^2+6x-4\right)=-\left(x^2+6x+9-13\right)\)

\(=-\left[\left(x+3\right)^2-13\right]=-\left(x+3\right)^2+13\le13\)

Vậy \(A_{max}=13\Leftrightarrow x+3=0\Leftrightarrow x=-3\)

\(B=3x^2-6x+1=\left(\sqrt{3}x\right)^2-2.\sqrt{3}x.\sqrt{3}+3-2\)

\(=\left(\sqrt{3}x-\sqrt{3}\right)^2-2\ge-2\)

Vậy \(B_{min}=-2\Leftrightarrow\sqrt{3}x-\sqrt{3}=0\Leftrightarrow x=1\)

\(C=5x^2-2x-3=\left(\sqrt{5}x\right)^2-2.\sqrt{5}x.\frac{1}{\sqrt{5}}+\frac{1}{5}-\frac{16}{5}\)

\(=\left(\sqrt{5}x-\frac{1}{\sqrt{5}}\right)^2-\frac{16}{5}\ge-\frac{16}{5}\)

Vậy \(C_{min}=-\frac{16}{5}\Leftrightarrow\sqrt{5}x-\frac{1}{\sqrt{5}}=0\Leftrightarrow\sqrt{5}x=\frac{1}{\sqrt{5}}\Leftrightarrow x=\frac{1}{5}\)

Với \(x\ge\dfrac{1}{3}\Leftrightarrow B=9x^2-6x-4\left(3x-1\right)+6=9x^2-18x+10\)

\(B=9\left(x^2-2x+1\right)+1=9\left(x-1\right)^2+1\ge1\\ B_{min}=1\Leftrightarrow x=1\left(1\right)\)

Với \(x< \dfrac{1}{3}\Leftrightarrow B=9x^2-6x+4\left(3x-1\right)+6=9x^2+6x+2\)

\(B=\left(9x^2+6x+1\right)+1=\left(3x+1\right)^2+1\ge1\\ B_{min}=1\Leftrightarrow x=-\dfrac{1}{3}\left(2\right)\)

\(\left(1\right)\left(2\right)\Leftrightarrow B_{min}=1\Leftrightarrow\left[{}\begin{matrix}x=1\\x=-\dfrac{1}{3}\end{matrix}\right.\)

\(C=2x^2+4x+2+3x^2+12x+12-4x^2-24x-36\\ C=x^2-8x-22=\left(x^2-8x+16\right)-38=\left(x-4\right)^2-38\ge-38\\ C_{min}=-38\Leftrightarrow x=4\)

Với \(x\ge\dfrac{1}{6}\Leftrightarrow A=5x^2-6x+1-1=5x^2-6x\)

\(A=5\left(x^2-2\cdot\dfrac{3}{5}x+\dfrac{9}{25}\right)-\dfrac{9}{5}=5\left(x-\dfrac{3}{5}\right)^2-\dfrac{9}{5}\ge-\dfrac{9}{5}\\ A_{min}=-\dfrac{9}{5}\Leftrightarrow x=\dfrac{3}{5}\left(1\right)\)

Với \(x< \dfrac{1}{6}\Leftrightarrow A=5x^2+6x-1-1=5x^2+6x-2\)

\(A=5\left(x^2+2\cdot\dfrac{3}{5}x+\dfrac{9}{25}\right)-\dfrac{19}{5}=5\left(x+\dfrac{3}{5}\right)^2-\dfrac{19}{5}\ge-\dfrac{19}{5}\\ A_{min}=-\dfrac{19}{5}\Leftrightarrow x=-\dfrac{3}{5}\left(2\right)\\ \left(1\right)\left(2\right)\Leftrightarrow A_{min}=-\dfrac{19}{5}\Leftrightarrow x=-\dfrac{3}{5}\)