Question

已知f(x)是偶函数,当x≥0时,f(x)=a^x(a>1),若不等式f(x)≤4的解集为[-2,2],求a

Answer 1

when x < 0
-x >0

f(x) = f(-x)
= a^(-x)

f(x) = a^x , x > 0
= 1 , x=0
= a^(-x) , x<0

for x > 0
f(x) ≤4
a^(x) ≤4
x ≤ log(a) 4

for x< 0
f(x) ≤4
a^(-x) ≤4
-x ≤ log(a) 4
ie
-log(a) 4 ≤x≤ log(a) 4

ie log(a) 4 = 2
4 = a^2
a = 2

Answer 2

由题意知，当x<=0时，f(x)=(1/a)^x
a^2=2，得a=2。
将a=2代入原函数中验证不等式，结果成立，因此a=2