$A = x^2 + 2x + 5 = x^2 + 2x + 2^2 + 1 =(x+2)^2 + 1$
$(x+2)^2 \geq 0 ⇒ (x+2)^2 + 1 \geq 1 $ hay $A \geq 1$
Ta thấy A = 1 khi $x+2 = 0 ⇔ x = -2$
Vậy min A = 1 khi x = -2
$B = 4x^2 - 4x + 6 = (2x)^2 - 4x + 1 + 5 = (2x-1)^2 + 5$
$(2x -1)^2 \geq 0 ⇒ (2x-1)^2 + 5 \geq 5 $ hay $B \geq 5$
Ta thấy B = 5 khi $2x - 1 = 0 ⇔ x =\dfrac{1}{2}$
Vậy min B = 5 khi $x = \dfrac{1}{2}$
$C = x^2 - 6x + 20 = x^2 - 6x + 9 + 11 = (x-3)^2 + 11$
$(x-3)^2 \geq 0 ⇒ (x-3)^2 + 11 \geq 11$ hay $C \geq 11$
Ta thấy C = 11 khi $x- 3 = 0 ⇔ x = 3$
Vậy min C = 11 khi x = 3
$E = 2x^2 - 4x + 10 = 2x^2 - 4x + 8 + 2 = 2(x^2 - 4x + 4) + 8 = 2(x-2)^2 + 8$
$2(x-2)^2 \geq 0 ⇒ 2(x-2)^2 + 8 \geq 8$ hay $E \geq 8$
Ta thấy E = 8 khi $x-2 = 0 ⇔ x = 2$
Vậy min E = 8 khi x = 2
$F = x^2 - x + 1 = x^2 - 2.x.0.5 + (0.5)^2 + 0.75 = (x-0.5)^2 + 0.75$
$(x-0.5)^2 \geq 0 ⇒ (x-0.5)^2 + 0.75 \geq 0.75$ hay $F \geq 0.75$
Ta thấy F = 0.75 khi $x - 0.5 = 0⇔ x = 0.5$
Vậy min F = 0.75 khi x = 0.5
$G = x^2 - 3x + 5 = x^2 - 3x + (1.5)^2 + 2.75 = (x-1.5)^2 + 2.75$
$(x-1.5)^2 \geq 0 ⇒(x-1.5)^2 + 2.75 \geq 2.75$ hay $G \geq 2.75$
Ta thấy G = 2.75 khi $x-1.5 = 0 ⇔ x = 1.5$
Vậy min G = 2.75 khi x = 1.5
$H = 2x^2 - 3x + 4 = 2x^2 - 3x + 1.125 + 2.875 = 2(x^2 - 1.5x + 0.5625) + 2.875 = 2(x-0.75)^2+2.875$
$2(x-0.75)^2 \geq 0 ⇒ 2(x-0.75)^2+2.875 \geq 2.875$ hay $H \geq 2.875$
Ta thấy H = 2.875 khi $x - 0.75 = 0 ⇔ x = 0.75$
Vậy min H = 2.875 khi x = 0.75
$K = (x-1)^2 + (x-4)^2 + 5$
$ =x^2 - 2x + 1 + x^2 - 8x + 16 + 5$
$=2x^2 - 10x + 22$
$=2x^2 - 10x + 12.5 + 9.5$
$=2(x^2 - 5x + 6.25) + 9.5$
$=2(x-2.5)^2 + 9.5$
$2(x-2.5)^2 \geq 0 ⇒ 2(x-2.5)^2 + 9.5 \geq 9.5$ hay $K \geq 9.5$
Ta thấy K = 9.5 khi $x - 2.5 = 0 ⇔ x = 2.5$
Vậy min K = 9.5 khi x = 2.5
