`4)`
`H = (x+1)(x-2)(x-3)(x-6)`
` = [ (x+1)(x-6)] . [(x-2)(x-3)]`
`= (x^2 - 6x + x - 6)(x^2 - 3x - 2x + 6)`
` = (x^2 - 5x-6)(x^2 - 5x + 6)`
` = [ (x^2 - 5x) - 6] . [ (x^2 - 5x) + 6]`
`= (x^2- 5x)^2 - 6^2`
`= (x^2 - 5x)^2 - 36`
`\forall x` ta có :
`(x^2- 5x)^2 \ge 0`
`=> (x^2 - 5x)^2 - 36 \ge -36`
`=> H \ge -36`
Dấu `=` xảy ra `<=> x^2 - 5x=0`
`<=> x(x-5)=0`
`<=>x = 5` hoặc `x=0`
Vậy `\text{Min}_H = -36 <=> x \in {0;5}`
`5)`
`F = (x-8)^2 + (x+7)^2`
`= (x^2 - 2 . x . 8 + 8^2) + (x^2 + 2 . x . 7 +7^2)`
` = (x^2 - 16x + 64) + (x^2 + 14x + 49)`
` = x^2 - 16x + 64 + x^2 + 14x + 49`
` = (x^2 +x^2) + (14x - 16x) + (64 +49)`
` = 2x^2 - 2x + 113`
` = 2 (x^2 - x + 1/4) + 225/2`
`= 2 [ x^2 - 2 . x . 1/2 + (1/2)^2] + 225/2`
` = 2 (x-1/2)^2+ 225/2`
`\forall x` ta có :
`(x-1/2)^2 \ge 0`
`=> 2 (x-1/2)^2 \ge 0`
`=> 2 (x-1/2)^2 + 225/2 \ge 225/2`
`=> F \ge 225/2`
Dấu `=` xảy ra `<=>x - 1/2=0`
`<=>x=1/2`
Vậy `\text{Min}_F= 225/2 <=> x=1/2`
