Question

[Nguyễn Mary]

tìm GTNN:

a, \(A=\dfrac{4x^2-6x+1}{\left(2x-1\right)^2}\)

b, \(B=\dfrac{3x^2-2x+3}{x^2+1}\)

Answer 1

[Trần Quốc Lộc]

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

Do \(\dfrac{\left(x-1\right)^2}{x^2+1}\ge0\forall x\)

\(\Rightarrow B=\dfrac{\left(x-1\right)^2}{x^2+1}+2\ge2\forall x\)

Dấu "=" xảy ra khi :

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

Vậy \(B_{\left(Min\right)}=2\) khi \(x=1\)

Answer 2

[Trần Quốc Lộc]

\(A=\dfrac{4x^2-6x+1}{\left(2x-1\right)^2}=\dfrac{4x^2-4x-2x+1+1-1}{\left(2x-1\right)^2}\\ =\dfrac{\left(4x^2-4x+1\right)-\left(2x-1\right)-1}{\left(2x-1\right)^2}\\ =\dfrac{\left(2x-1\right)^2}{\left(2x-1\right)^2}-\dfrac{2x-1}{\left(2x-1\right)^2}-\dfrac{1}{\left(2x-1\right)^2}\\ =1-\dfrac{1}{2x-1}-\dfrac{1}{\left(2x-1\right)^2}\)

Đặt \(-\dfrac{1}{2x-1}=y\)

\(\Rightarrow A=1+y+y^2\\ =y^2+y+\dfrac{1}{4}+\dfrac{3}{4}\\ =\left(y^2+y+\dfrac{1}{4}\right)+\dfrac{3}{4}\\ =\left(y+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\)

Do \(\left(y+\dfrac{1}{2}\right)^2\ge0\forall x\)

\(\Rightarrow\left(y+\dfrac{1}{2}\right)^2+\dfrac{3}{4}\ge\dfrac{3}{4}\forall x\)

Dấu "=" xảy ra khi:

\(\left(y+\dfrac{1}{2}\right)^2=0\\ \Leftrightarrow y+\dfrac{1}{2}=0\\ \Leftrightarrow y=-\dfrac{1}{2}\\ \Leftrightarrow-\dfrac{1}{2x-1}=-\dfrac{1}{2}\\ \Leftrightarrow2x-1=2\\ \Leftrightarrow2x=3\\ \Leftrightarrow x=\dfrac{3}{2}\)

Vậy \(A_{\left(Min\right)}=\dfrac{3}{4}\) khi \(x=\dfrac{3}{2}\)