\(x\left(2x-3\right)=2x^2-3x=2\left(x^2-\dfrac{3}{2}x+\dfrac{9}{16}\right)-\dfrac{9}{8}\\ =2\left(x-\dfrac{3}{4}\right)^2-\dfrac{9}{8}\)
Với mọi x thì \(2\left(x-\dfrac{3}{4}\right)^2\ge0\)
=>\(2\left(x-\dfrac{3}{4}\right)^2-\dfrac{9}{8}\ge-\dfrac{9}{8}\)
Dấu ''='' xảy ra khi:\(\left(x-\dfrac{3}{4}\right)^2=0\)
\(\Leftrightarrow x-\dfrac{3}{4}=0\)
\(\Leftrightarrow x=\dfrac{3}{4}\)
Vậy...
\(A=x\left(2x-3\right)\)
\(\Leftrightarrow A=2x^2-3x\)
\(\Leftrightarrow A=2x^2-3x+\dfrac{9}{8}-\dfrac{9}{8}\)
\(\Leftrightarrow A=2\left(x^2-\dfrac{3}{2}x+\dfrac{9}{16}\right)-\dfrac{9}{8}\)
\(\Leftrightarrow A=2\left[x^2-2.x.\dfrac{3}{4}+\left(\dfrac{3}{4}\right)^2\right]-\dfrac{9}{8}\)
\(\Leftrightarrow A=2\left(x-\dfrac{3}{4}\right)^2-\dfrac{9}{8}\)
Vậy GTNN của \(A=\dfrac{-9}{8}\) khi \(x-\dfrac{3}{4}=0\Leftrightarrow x=\dfrac{3}{4}\)
A = x( 2x - 3)
A = 2x2 - 3x
A = 2(\(x^2-\dfrac{3}{2}x\))
A =2[ x2 - 2.\(\dfrac{3}{4}\)x + \(\left(\dfrac{3}{4}\right)^2\)] + 2.\(\dfrac{9}{16}\)
A = 2( x - \(\dfrac{3}{4}\))2 + \(\dfrac{9}{8}\)
Do : 2( x - \(\dfrac{3}{4}\))2 lớn hơn hoặc bằng 0 với mọi x
Suy ra : 2( x - \(\dfrac{3}{4}\))2 + \(\dfrac{9}{8}\)lớn hơn hoặc bằng \(\dfrac{9}{8}\) với mọi x
Vậy , Amin = \(\dfrac{9}{8}\)khi và chỉ khi x - \(\dfrac{3}{4}\)= 0 -> x = \(\dfrac{3}{4}\)
Nhầm sử cho tớ thành - \(\dfrac{9}{8}\)nha
\(x\left(2x-3\right)\\ =2x^2-3x\\ =2x^2-3x+\dfrac{9}{8}-\dfrac{9}{8}\\ =\left(2x^2-3x+\dfrac{9}{8}\right)-\dfrac{9}{8}\\ =2\left(x^2-\dfrac{3}{2}x+\dfrac{9}{16}\right)-\dfrac{9}{8}\\ =2\left[x^2-2\cdot x\cdot\dfrac{3}{4}+\left(\dfrac{3}{4}\right)^2\right]-\dfrac{9}{8}\\ =2\left(x-\dfrac{3}{4}\right)^2-\dfrac{9}{8}\\ Do\text{ }\left(x-\dfrac{3}{4}\right)^2\ge0\forall x\\ \Rightarrow2\left(x-\dfrac{3}{4}\right)^2\ge0\forall x\\ \Rightarrow2\left(x-\dfrac{3}{4}\right)^2-\dfrac{9}{8}\ge-\dfrac{9}{8}\forall x\\ \text{Dấu “=” xảy ra khi : }\\ \left(x-\dfrac{3}{4}\right)^2=0\\ \Leftrightarrow x-\dfrac{3}{4}=0\\ \Leftrightarrow x=\dfrac{3}{4}\\ \text{Vậy }GTNN\text{ }\text{của biểu thức là }-\dfrac{9}{8}\text{ }khi\text{ }x=\dfrac{3}{4}\)
