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
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[sqrt(a+1/2)+sqrt(b+1/2)]^2
=a+1/2+b+1/2+2*sqrt(ab+1/2*(a+b)+1/4)
=2+2*sqrt(ab+3/4)
because
0<=ab<=[(a+b)/2]^2=1/4
so
sqrt[2+2*sqrt(0+3/4)] <= sqrt(a+1/2)+sqrt(b+1/2) <= sqrt[2+2*sqrt(1/4+3/4)]
sqrt[2+sqrt(3)] <= sqrt(a+1/2)+sqrt(b+1/2) <= 2
=a+1/2+b+1/2+2*sqrt(ab+1/2*(a+b)+1/4)
=2+2*sqrt(ab+3/4)
because
0<=ab<=[(a+b)/2]^2=1/4
so
sqrt[2+2*sqrt(0+3/4)] <= sqrt(a+1/2)+sqrt(b+1/2) <= sqrt[2+2*sqrt(1/4+3/4)]
sqrt[2+sqrt(3)] <= sqrt(a+1/2)+sqrt(b+1/2) <= 2
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