已知f（x）=x^2+2（a-2）x+41.如果对一切x∈R,f（x）＞0恒成立,求实数a的取值范围.2.如果对x∈[1,3],f（x）＞0恒成立,求实数a的取值范围

Question

已知f（x）=x^2+2（a-2）x+4
1.如果对一切x∈R,f（x）＞0恒成立,求实数a的取值范围.2.如果对x∈[1,3],f（x）＞0恒成立,求实数a的取值范围

Answer 1

f(x)为开口向上的抛物线,对称轴为x= 2 - a, 顶点(2 - a, 4 - (a - 2)²)
(1) 如果对一切x∈R,f(x) ＞0恒成立, 则4 - (a - 2)² > 0, a(a - 4) < 0, 0 < a < 4
(2)
(i) 对称轴x = 2 - a ∈[1, 3], 1 ≤ 2 - a ≤ 3, -1 ≤ a ≤ 1
4 - (a - 2)² > 0, 0 < a < 4
结合前提: 0 < a ≤ 1
(ii) 对称轴x = 2 - a < 1, a > 1
此时[1,3]在对称轴右侧,f(x) > 0, 只须f(1) > 0
f(1) = 1 + 2(a - 2) + 4 = 2a + 1 >0
a > -1/2
结合前提, a > 1
(iii)对称轴x = 2 - a > 3, a < -1
此时[1,3]在对称轴左侧, f(x) > 0, 只须f(3) > 0
f(3) = 9 + 6(a - 2) + 4 = 1 + 6a > 0
a > 1/6
结合前提: -1/2 < a < 1
与a < -1矛盾,此时无解.
(i)(ii)(iii)结合: a > 0