Đáp án:
`a)`
`M=x^2+6x+10`
`=x^2+6x+9+1`
`=x^2+2.3.x+3^2+1`
`=(x+3)^2+1`
Vì `(x+3)^2>=0∀x`
`->(x+3)^2+1≥1∀x`
`->M≥1`
Dấu `'='` xảy ra `<=>x+3=0<=>x=-3`
Vậy `M_{min}=1` khi `x=-3`
`b)`
`N=2x^2+4x`
`=2x^2+4x+2-2`
`=2.(x^2+2x+1)-2`
`=2.(x+1)^2-2`
Vì `(x+1)^2≥0∀x`
`->2.(x+1)^2≥0∀x`
`->2.(x+1)^2-2≥-2∀x`
`->N≥-2`
Dấu `'='` xảy ra `<=>x+1=0<=>x=-1`
Vậy `N_{min}=-2` khi `x=-1`
`c)`
`P=-x^2-2x+9`
`=-x^2-2x-1+10`
`=-(x^2+2x+1)+10`
`=-(x+1)^2+10`
Vì `(x+1)^2≥0∀x`
`->-(x+1)^2≤0∀x`
`->-(x+1)^2+10≤10∀x`
`->P≤10`
Dấu `'='` xảy ra `<=>x+1=0<=>x=-1`
Vậy `P_{max}=10` khi `x=-1`
`d)`
`Q=3-2x^2-8x`
`=-2x^2-8x-8+11`
`=-2.(x^2+4x+4)+11`
`=-2.(x+2)^2+11`
Vì `(x+2)^2≥0∀x`
`->-2(x+2)^2≤0∀x`
`->-2(x+2)^2+11≤11∀x`
`->Q≤11`
Dấu `'='` xảy ra `<=>x+2=0<=>x=-2`
Vậy `Q_{max}=11` khi `x=-2`
