Question

[Du Bách Lý]

tim GTNN cua bt

a) \(x^4+x^2+2\)

b)\(3x^2-21x+15\)

c)\(x^2-4xy+5y^2+10x-22y+28\)

Answer 1

[Phùng Khánh Linh]

a) A = x⁴ + x² + 2

Do : x⁴ ≥ 0 ∀x

x² ≥ 0 ∀x

⇒ x⁴ + x² + 2 ≥ 2

⇒ A_Min = 2 ⇔ x = 0

b) B = 3x² - 21x + 15

B = 3( x² - \(2\dfrac{7}{2}x+\dfrac{49}{4}\) ) + 15 - \(\dfrac{147}{4}\)

B = 3( x - \(\dfrac{7}{2}\))² - \(\dfrac{87}{4}\)

Do : 3( x - \(\dfrac{7}{2}\))² ≥ 0 ∀x

⇒ 3( x - \(\dfrac{7}{2}\))² - \(\dfrac{87}{4}\) ≥ - \(\dfrac{87}{4}\)

⇒ B_Min = - \(\dfrac{87}{4}\) ⇔ x = \(\dfrac{7}{2}\)

c) C = x² - 4xy + 5y² + 10x - 22y + 28

C = x² - 4xy + 4y² + 10x - 20y + 25 + y² - 2y + 1 + 2

C = ( x - 2y)² + 10( x - 2y) + 25 + ( y - 1)² + 2

C = ( x - 2y + 5)² + ( y - 1)² + 2

Do : ( x - 2y + 5)² ≥ 0 ∀xy

( y - 1)² ≥ 0 ∀y

⇒ ( x - 2y + 5)² + ( y - 1)² + 2 ≥ 2

⇒ C_Min = 2 ⇔ x = - 3 ; y = 1