Lời giải:
a) `A=4x-x^2+3`
`=-(x^2-4x-3)`
`=-[(x-2)^2-7]`
`=-(x-2)^2+7`
Vì `-(x-2)^2≤0∀x`
`⇒-(x-2)^2+7≤7∀x`
`⇒A_max=7⇔x=2`
b) `B=x-x^2`
`=-(x^2-x)`
`=-[(x-\frac{1}{2})^2-\frac{1}{4}]`
`=-(x-\frac{1}{2})^2+\frac{1}{4}`
Vì `-(x-\frac{1}{2})^2≤0∀x`
`⇒-(x-\frac{1}{2})^2+\frac{1}{4}≤\frac{1}{4}∀x`
`⇒B_max=\frac{1}{4}⇔x=\frac{1}{2}`
c) `N=2x-2x^2-5`
`=-(2x^2-2x+5)`
`=-2[(x-\frac{1}{2})^2+\frac{9}{4}]`
`=-2(x-\frac{1}{2})^2-\frac{9}{2}`
Vì `-2(x-\frac{1}{2})^2≤0∀x`
`⇒-2(x-\frac{1}{2})^2-\frac{9}{2}≤-\frac{9}{2}∀x`
`⇒N_max=-\frac{9}{2}⇔x=\frac{1}{2}`
