tìm GTLN E = 2x^2 + 4x + 9 / x^2 +2x + 4

$\\$

`E = (2x^2 + 4x + 9)/(x^2 +2x+4)`

`->E =(2x^2 + 4x + 2 + 7)/(x^2 + 2 . x . 1 + 1^2 +3)`

`->E = (2 (x^2 + 2x + 1) + 7)/( (x+1)^2 + 3)`

`->E = (2 (x+ 1)^2 + 7)/( (x+1)^2 + 3)`

`->E = (2 [(x+1)^2 + 3] + 1)/( (x+1)^2 + 3)`

`->E = (2 [(x+1)^2 +3])/( (x+1)^2 +3) + 1/( (x+1)^2 + 3)`

`->E=2 + 1/( (x+1)^2 + 3)`

Vì `(x+1)^2 ≥0∀x`

`-> (x+1)^2 + 3 ≥3∀x`

`-> 1/( (x+1)^2 +3) ≤ 1/3 ∀x`

`->2 + 1/((x+1)^2 +3) ≤2 + 1/3=7/3 ∀x`

`->E ≤ 7/3 ∀x`

Dấu "`=`" xảy ra khi :

`(x+1)^2=0 ↔ x+1=0 ↔x=-1`

Vậy `max E=7/3 ↔x=-1`