Question

已知a,b都大于等于0,a+b=1,则根号(a+1/2)+根号(b+1/2)的范围是

Answer 1

设根号(a+1/2)=x 根号(b+1/2)=y
a=x^2-1/2 b=y^2-1/2
a+b=x^2+y^2-1=1
x^2+y^2=2 2xy<=x^2+y^2=2
x^2+y^2+2xy<=4
(x+y)^2<=4 0<x+y<=2
根号(a+1/2)+根号(b+1/2)的范围是(0,2]

Answer 2

[√(a+1/2)+√(b+1/2)]²
=a+1/2+b+1/2+2√[(a+1/2)(b+1/2)]
=1+1+2√(ab+1/2(a+b)+1/4)

1=a+b>=2√ab
√ab<=1/2
ab<=1/4

ab+1/2(a+b)+1/4
=ab+3/4<=1/4+3/4=1
所以[√(a+1/2)+√(b+1/2)]²<=1+1+2√1=4
所以0<√(a+1/2)+√(b+1/2)≤2

Answer 3

a+b=1,b=1-a
设y＝√(a+1/2)+√(b+1/2)=√(a+1/2)+√(1-a+1/2)=√(a+1/2)+√(3/2-a)
则y^2＝(a+1/2)+(3/2-a)+2√(a+1/2)(3/2-a)=2+2√-a^2+a+3/4=2+2√[-(a+1/2)^2+1]
a=-1/2 有最大值 此时y＝2
a＝1/2有最小值 y＝√2