Question

Tìm giá trị nhỏ nhất

1) y = x² + 5x - 4

2) y = 2x² - 6x + 5

3) y = x² - 4x + 1

4) 2x² - 8x + 3

5) y = \(\frac{2x+1}{x^2}\)

6) y = \(\frac{4x^2-2x+1}{x^2}\)

Answer 1

1: \(y=x^2+2\cdot x\cdot\dfrac{5}{2}+\dfrac{25}{4}-\dfrac{41}{4}\)

\(=\left(x+\dfrac{5}{2}\right)^2-\dfrac{41}{4}\ge-\dfrac{41}{4}\forall x\)

Dấu '=' xảy ra khi x=-5/2

2: \(y=2\left(x^2-2x+\dfrac{5}{2}\right)\)

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

\(=2\left(x-1\right)^2+3\ge3\forall x\)

Dấu '=' xảy ra khi x=1

3: \(y=x^2-4x+4-3=\left(x-2\right)^2-3\ge-3\forall x\)

Dấu '=' xảy ra khi x=2

4: \(2x^2-8x+3\)

\(=2\left(x^2-4x+\dfrac{3}{2}\right)\)

\(=2\left(x^2-4x+4-\dfrac{5}{2}\right)\)

\(=2\left(x-2\right)^2-5\ge-5\forall x\)

Dấu '=' xảy ra khi x=2