已知函数f(x)=x^2-4ax+2a+6(a∈R)(1)若函数的值域为[0,+∞)求a的值;(2)若函数的值域为非负数,求函数f(a)=2-a∣a+3∣的值域 速求!
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![已知函数f(x)=x^2-4ax+2a+6(a∈R)(1)若函数的值域为[0,+∞)求a的值;(2)若函数的值域为非负数,求函数f(a)=2-a∣a+3∣的值域 速求!](/uploads/image/z/14107334-14-4.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3Dx%5E2-4ax%2B2a%2B6%28a%E2%88%88R%29%281%29%E8%8B%A5%E5%87%BD%E6%95%B0%E7%9A%84%E5%80%BC%E5%9F%9F%E4%B8%BA%5B0%2C%2B%E2%88%9E%EF%BC%89%E6%B1%82a%E7%9A%84%E5%80%BC%EF%BC%9B%EF%BC%882%EF%BC%89%E8%8B%A5%E5%87%BD%E6%95%B0%E7%9A%84%E5%80%BC%E5%9F%9F%E4%B8%BA%E9%9D%9E%E8%B4%9F%E6%95%B0%2C%E6%B1%82%E5%87%BD%E6%95%B0f%28a%29%3D2-a%E2%88%A3a%2B3%E2%88%A3%E7%9A%84%E5%80%BC%E5%9F%9F+%E9%80%9F%E6%B1%82%21)
已知函数f(x)=x^2-4ax+2a+6(a∈R)(1)若函数的值域为[0,+∞)求a的值;(2)若函数的值域为非负数,求函数f(a)=2-a∣a+3∣的值域 速求!
已知函数f(x)=x^2-4ax+2a+6(a∈R)
(1)若函数的值域为[0,+∞)求a的值;(2)若函数的值域为非负数,求函数f(a)=2-a∣a+3∣的值域
速求!
已知函数f(x)=x^2-4ax+2a+6(a∈R)(1)若函数的值域为[0,+∞)求a的值;(2)若函数的值域为非负数,求函数f(a)=2-a∣a+3∣的值域 速求!
(1)f(x)=(x-2a)^2-4a^2+2a+6
-4a^2+2a+6=0
a=-1或a=3/2
(2)y=x^2-4ax+2a+6 是开口向上的抛物线
其值始终为非负数,所以最小值 为非负数
y = x^2 - 2 * 2a * x + (2a)^2 - (2a)^2 + 2a + 6
= (x - 2a)^2 - 2(2a^2 -a -3)
最小值为 -2(2a^2 - a -3)
-2 (2a^2 - a -3) ≥ 0
(2a -3)(a + 1) ≤0
-1 ≤ a ≤ 3/2
在此范围内 a + 3 恒大于0
f(a) = 2 - a(a+3)
= -a^2 - 3a + 2
= - [a^2 + 3a -2]
= - [(a + 3/2)^2 - 9/4 - 2]
= 17/4 - (a + 3/2)^2
f(a) 是 以 a = -3/2 为顶点的抛物线.在顶点两侧单调递减.
区间 -1 ≤ a ≤ 3/2 在 a = -3/2 右侧.
最是值为
f(-1) = 2 - (-1)*|-1 + 3| = 4
最小值为
f(3/2) = 2 - (3/2)|3/2 + 3| = -19/4
(1)
f(x)=x^2-4ax+2a+6
=(x-2a)^2-4a^2+2a+6
>=-4a^2+2a+6>=0
-2a^2+a+3>=0
2a^2-a-3<=0
(2a-3)(a+1)<=0
-1<=a<=3/2
(2)
-1<=a<=3/2
f(a)=2-a∣a+3∣
=2-a(a+3)
=-a^2-3a+2
=-(a+(3/2))^2+(17/4)
因1/2<=a+(3/2)<=3
所以:
-9+(17/4)<=f(a)<=-(1/4)+(17/4)
-19/4<=f(a)<=4