f(x)=a(1-x^2)^1/2+(1+x)^1/2+(1-x)^1/2

Ⅰ t∈〔√2,2〕
m(t)=a(1/2t^2-1)+t=1/2at^2+t-a
Ⅱ x=-1/a为g(a)对称轴,
⑴a＞0时,y=m(t),g(a)=m(2)=a+2;
⑵a=0时,m(t）=t,t∈〔√2,2〕,g(a)=2;
⑶a＜0时,y=m(t),t∈〔√2,2〕,的图像是开口向下的抛物线的一段,
①若t=-1/a∈(0,√2〕即a≤-√2/2时,g(a)=m(√2）=√2;
② 若t=-1/a∈〔√2,2〕即a∈〔√2/2,-1/2〕时,g(a)=m(1/a)=-a-1/2a;
③若t=-1/a∈(2,+∞)即a∈(-1/2,0)时,g(a)=m(2)=a+2.
综上,有…
Ⅲ 当a＞-1/2时,g(a)=a+2＞3/2＞√2;
当-√2/2≤a≤-1/2时,-a∈〔1/2,√2/2),-1/2a∈(√2/2,1〕,∴-a≠-1/2a,g(a)=-a-1/2a＞2√〔（-a）*（-1/2a）〕=√2,故当a＞-√2/2时,g(a)＞√2;
当a＞0时,1/a＞0,由g(a)=g(1/a)知:a+2=1/a+2,故a=1;
当a＜0时,a*1/a=1,故a≤-1或1/a≤-1,从而有g(a)=√2或g(1/a)=√2,要使g(a)=g(1/a),必须有a≤-√2/2,1/a≤-√2/2,即-√2≤a≤-√2/2,此时,g(a)=√2=g(1/a).
综上,满足条件的所有实数a为：-√2≤a≤-√2/2或a=1