Question

若实数a和b满足2*4^n-2^a*3^b+2*9^b=2^a+3^b+1,则2^a+3^b的取值范围

Answer 1

设x=2^a>0,y=3^b>0,x+y=u>0,由2*4^a-2^a*3^b+2*9^b=2^a+3^b+1(改题了）得
2u^2-5xy=u+1,
0<xy<=(x+y)^2/4=u^2/4,
∴2u^2-5u^2/4<=u+1,且2u^2>u+1,
∴（3/4）u^2-u-1<=0,且2u^2-u-1>0,
∴-2/3<=u<=2,且u>1,
解得1<u<=2,为所求.

追问

好棒！

追答

过奖了。