f(x)=1-2a-2acosx-2sin^2x的最小值为g(a).(1)求g(a).(2)求能使g(a)=1/2的a值,并求当a取此值时f(x)的最大值
来源:学生作业帮助网 编辑:作业帮 时间:2024/07/12 06:10:06
![f(x)=1-2a-2acosx-2sin^2x的最小值为g(a).(1)求g(a).(2)求能使g(a)=1/2的a值,并求当a取此值时f(x)的最大值](/uploads/image/z/638430-6-0.jpg?t=f%28x%29%3D1-2a-2acosx-2sin%5E2x%E7%9A%84%E6%9C%80%E5%B0%8F%E5%80%BC%E4%B8%BAg%28a%29.%281%29%E6%B1%82g%28a%29.%282%29%E6%B1%82%E8%83%BD%E4%BD%BFg%28a%29%3D1%2F2%E7%9A%84a%E5%80%BC%2C%E5%B9%B6%E6%B1%82%E5%BD%93a%E5%8F%96%E6%AD%A4%E5%80%BC%E6%97%B6f%28x%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC)
f(x)=1-2a-2acosx-2sin^2x的最小值为g(a).(1)求g(a).(2)求能使g(a)=1/2的a值,并求当a取此值时f(x)的最大值
f(x)=1-2a-2acosx-2sin^2x的最小值为g(a).(1)求g(a).
(2)求能使g(a)=1/2的a值,并求当a取此值时f(x)的最大值
f(x)=1-2a-2acosx-2sin^2x的最小值为g(a).(1)求g(a).(2)求能使g(a)=1/2的a值,并求当a取此值时f(x)的最大值
f(x) = 1 - 2a - 2a cosx - 2sin²x
= 1 - 2a - 2a cosx - 2(1 - cos²x)
= 1 - 2a - 2a cosx - 2 + 2cos²x
= 2cos²x - 2a cosx - 2a - 1
= 2(cos²x - acosx) - 2a - 1
= 2[cos²x - 2(cosx)(a/2) + (a/2)² - (a/2)²] - 2a - 1
= 2(cosx - a/2)² - 2(a/2)² - 2a - 1
= 2(cosx - a/2)² - a²/2 - 2a - 1
抛物线开口向上,有最小值g(a) = -a²/2 - 2a - 1
-----------------------------------------------------------------------------------------------------------------------
g(a) = 1/2
-a²/2 - 2a - 1 = 1/2
a² + 4a + 3 = 0
a = -1 或 a = -3
-----------------------------------------------------------------------------------------------------------------------
当a = -1时
f(x) = 2(cosx + 1/2)² + 1/2
最大值 = 2(1 + 1/2)² + 1/2 = 5
当a = -3时
f(x) = 2(cosx + 3/2)² + 1/2
最大值 = 2(1 + 3/2)² + 1/2 = 13