若函数f(x)=1+a(1/2)^x+(1/4)^x在「0,+∞)上是以3为上界的有界实数,则实数a的取值范围是

Question

若函数f(x)=1+a(1/2)^x+(1/4)^x在「0,+∞)上是以3为上界的有界实数,则实数a的取值范围是

Answer 1

属于(0,1],y=1+at+t^2,对称轴为- a/2
若-a/2>=1（a<=-2),y(t)在（0,1]单调递减,ymax=y(0)=1,ymin=y(1)=2+a,由题,|2+a|<=3,所以,-5<=a<=-2;
若-a/2<=0(a>=0),y(t)在（0,1]单调递增,ymin=y(0)=1,ymax=y(1)=2+a,由题,|2+a|<=3,所以,0<=a<=1;
若0综上,-5<=a<=1