\(A=x^2-6x+10\)

\(\Leftrightarrow A=x^2-2\cdot x\cdot3+3^2-9+10\)

\(\Leftrightarrow A=\left(x-3\right)^2+1\ge1\)     \(\forall x\in z\)

\(\Leftrightarrow A_{min}=1khix=3\)

\(B=3x^2-12x+1\)

\(\Leftrightarrow B=\left(\sqrt{3}x\right)^2-2\cdot\sqrt{3}x\cdot2\sqrt{3}+\left(2\sqrt{3}\right)^2-12+1\)

\(\Leftrightarrow B=\left(\sqrt{3}x-2\sqrt{3}\right)^2-11\ge-11\)    \(\forall x\in z\)

\(\Leftrightarrow B_{min}=-11khix=2\)

\(A=\frac{4x^2-12x+15}{x^2-3x+3}=4+\frac{3}{x^2-3x+3}=4+\frac{3}{\left(x-\frac{3}{2}\right)^2+\frac{3}{4}}\le8\)

dau '=' xay ra khi \(x=\frac{3}{2}\)

\(B=\frac{4x^2-8x+12}{x^2-2x+5}=4-\frac{8}{x^2-2x+5}=4-\frac{8}{\left(x-1\right)^2+4}\le2\)

dau '=' xay ra khi \(x=1\)

\(Q=x^2\left(4-3x\right)=\dfrac{4}{9}.\dfrac{3}{2}x.\dfrac{3}{2}x\left(4-3x\right)\)

\(Q\le\dfrac{1}{27}.\dfrac{4}{9}.\left(\dfrac{3x}{2}+\dfrac{3x}{2}+4-3x\right)^3=\dfrac{256}{243}\)

\(Q_{maxx}=\dfrac{256}{243}\) khi \(\dfrac{3x}{2}=4-3x\Leftrightarrow x=\dfrac{8}{9}\)

1)ĐK:`4x^2-12x+9>0`

`<=>(2n-3)^2>0`

`<=>2n-3 ne 0`

`<=>n ne 3/2`

`d)x^2-x+1`

`=(x-1/2)^2+3/4>0AAx`

`=>` bt xd `AAx in RR`

e)ĐK:`x^2-8x+15>0`

`<=>x^2-3x-5x+15>0`

`<=>x(x-3)-5(x-3)>0`

`<=>(x-3)(x-5)>0`

`TH1:` \(\begin{cases}x-3>0\\x-5>0\\\end{cases}\)

`<=>` \(\begin{cases}x>3\\x>5\\\end{cases}\)

`<=>x>5`

`TH2:` \(\begin{cases}x-3<0\\x-5<0\\\end{cases}\)

`<=>` \(\begin{cases}x<3\\x<5\\\end{cases}\)

`<=>x<3`

f)ĐK:`3x^2-7x+20>0`

`<=>x^2-2x+1+2x^2-5x+19>0`

`<=>(x-1)^2+2(x-5/2)^2+13/2>0` luôn đúng

c) Để biểu thức \(\dfrac{1}{\sqrt{4x^2-12x+9}}\) có nghĩa thì \(4x^2-12x+9>0\)

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

\(\Leftrightarrow2x-3\ne0\)

\(\Leftrightarrow2x\ne3\)

hay \(x\ne\dfrac{3}{2}\)

d) Để biểu thức \(\dfrac{1}{\sqrt{x^2-x+1}}\) có nghĩa thì \(x^2-x+1>0\)

\(\Leftrightarrow x^2-2\cdot x\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{3}{4}>0\)

\(\Leftrightarrow\left(x-\dfrac{1}{2}\right)^2+\dfrac{3}{4}>0\)(luôn đúng)

e) Để biểu thức \(\dfrac{1}{\sqrt{x^2-8x+15}}\) có nghĩa thì \(x^2-8x+15>0\)

\(\Leftrightarrow\left(x-4\right)^2>1\)

\(\Leftrightarrow\left[{}\begin{matrix}x-4>1\\x-4< -1\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x>5\\x< 3\end{matrix}\right.\)

f) Để biểu thức \(\dfrac{1}{\sqrt{3x^2-7x+20}}\) có nghĩa thì \(3x^2-7x+20>0\)

\(\Leftrightarrow x^2-\dfrac{7}{3}x+\dfrac{20}{3}>0\)

\(\Leftrightarrow x^2-2\cdot x\cdot\dfrac{7}{6}+\dfrac{49}{36}+\dfrac{191}{36}>0\)

\(\Leftrightarrow\left(x-\dfrac{7}{6}\right)^2+\dfrac{191}{36}>0\)(luôn đúng)

Bài 1:

a: A=x^2-6x+10

=x^2-6x+9+1

=(x-3)^2+1>=1

Dấu = xảy ra khi x=3

b: \(B=3x^2-12x+1\)

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

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

=3(x-2)^2-11>=-11

Dấu = xảy ra khi x=2

Bạn tham khảo lời giải tại đây:

Tìm GTLN của biểu thức: \(A=\left(\dfrac{x^2}{x^2-3x 2} \dfrac{x^2}{x^2-5x 6}\right):\dfrac{x^4 x^2 1}{x^2-4x 3}\) - Hoc24