f'(x)=a^x*lna+2x-lna=(a^x-1)lna+2x
f(x)的定义域是[-1,1]
∵a>1
当x∈[0,1]
f'(x)>=0
x∈[-1,0]
f'(x)<0
∴f(x)在[-1,0])减,(0,1)增
x1,x2∈[-1,1]
∴f(x)最大值
f(1)=a+1-lna
f(-1)=1/a+1+lna
f(1)-f（-1）
=(a^2-1-2alna)/a
a^2-1-2alna恒>0
∴f(x)最大值=f(1)=a+1-lna
f(x)最小值=f(0)=1+0-0=1
|F(x2)-F(x1)|max
=a+1-lna-1
=a-lna<=e^2-2
设f(a)=a-lna
f'(a)=1-1/a=(a-1)/a
∵a>1
∴f(a)单增
a=e^2
a-lna=e^2-2
∴a的取值范围(1,e^2)