Question

Tìm min(GTNN) của :

A = 2x² + 6x 

B = x² - 2x + y² - 4y + 6

C = x² - 2xy + 6y² - 12x + 2y + 45

D = (x - 1) (x + 2) (x + 3) (x + 6)

Answer 1

A = 2x² + 6x = 2( x² + 3x + 9/4 ) - 9/2 = 2( x + 3/2 )² - 9/2 ≥ -9/2 ∀ x

Dấu "=" xảy ra khi x = -3/2

=> MinA = -9/2 <=> x = -3/2

B = x² - 2x + y² - 4y + 6 = ( x² - 2x + 1 ) + ( y² - 4y + 4 ) + 1 = ( x - 1 )² + ( y - 2 )² + 1 ≥ 1 ∀ x, y

Dấu "=" xảy ra khi x = 1 ; y = 2

=> MinB = 1 <=> x = 1 ; y = 2

C = x² - 2xy + 6y² - 12x + 2y + 45

= ( x² - 2xy + y² - 12x + 12y + 36 ) + ( 5y² - 10y + 5 ) + 4

= [ ( x² - 2xy + y² ) - ( 12x - 12y ) + 36 ] + 5( y² - 2y + 1 ) + 4

= [ ( x - y )² - 2( x - y ).6 + 6² ] + 5( y - 1 )² + 4

= ( x - y - 6 )² + 5( y - 1 )² + 4 ≥ 4 ∀ x, y

Dấu "=" xảy ra khi x = 7 ; y = 1

=> MinC = 4 <=> x = 7 ; y = 1

D = ( x - 1 )( x + 2 )( x + 3 )( x + 6 )

= [ ( x - 1 )( x + 6 ) ][ ( x + 2 )( x + 3 ) ]

= ( x² + 5x - 6 )( x² + 5x + 6 )

= ( x² + 5x )² - 36 ≥ -36 ∀ x

Dấu "=" xảy ra <=> x² + 5x = 0

                        <=> x( x + 5 ) = 0

                        <=> x = 0 hoặc x = -5

=> MinD = -36 <=> x = 0 hoặc x = -5