Câu hỏi của Du Bách Lý

Question

tim GTNN cua bt

a) $x^4+x^2+2$

b)$3x^2-21x+15$

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

Phùng Khánh Linh · Accepted Answer

a) A = x4 + x2 + 2

Do : x4 ≥ 0 ∀x

x2 ≥ 0 ∀x

⇒ x4 + x2 + 2 ≥ 2

⇒ AMin = 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 - $\dfrac{87}{4}$ ≥ - $\dfrac{87}{4}$

⇒ BMin = - $\dfrac{87}{4}$ ⇔ 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

⇒ CMin = 2 ⇔ x = - 3 ; y = 1Câu trả lời: a) A = x4 + x2 + 2