已知偶函数f(x)在(0,+∞)上为增函数,且f(2)=0,解不等式f[log2(x2+5x+4)]≥0.

问题描述:

已知偶函数f(x)在(0,+∞)上为增函数,且f(2)=0,解不等式f[log2(x2+5x+4)]≥0

∵不等式f[log2(x2+5x+4)]≥0.且f(2)=0,
f(log2(x2+5x+4))≥f(2)
∵偶函数f(x)在(0,+∞)上为增函数,
|log2(x2+5x+4)|≥2,
log2(x2+5x+4)≥2log2(x2+5x+4)≤−2
∴x2+5x+4≥22或0<x2+5x+4≤2-2
解得x≥0或x≤-5,或

x>−1或x<−4
−5−
10
2
≤x≤
−5+
10
2

∴原不等式的解集为{x|x≥0或x≤-5或−1<x≤
−5+
10
2
−5−
10
2
≤x<−4
}