已知函数f(x)=3sinx2+cosx2(Ⅰ)求f(x)的周期、对称中心、对称轴和单调递增区间;(Ⅱ)当x∈[0,π]时,求f(x)的值域.-数学

题目简介

已知函数f(x)=3sinx2+cosx2(Ⅰ)求f(x)的周期、对称中心、对称轴和单调递增区间;(Ⅱ)当x∈[0,π]时,求f(x)的值域.-数学

题目详情

已知函数f(x)=
3
sin
x
2
+cos
x
2

(Ⅰ) 求f(x)的周期、对称中心、对称轴和单调递增区间;
(Ⅱ) 当x∈[0,π]时,求f(x)的值域.
题型:解答题难度:中档来源:不详

答案

(I)f(x)=2(
3
2
sinclass="stub"x
2
+class="stub"1
2
cosclass="stub"x
2
)=2sin(class="stub"x
2
+class="stub"π
6

∴T=class="stub"2π
class="stub"1
2
=4π
class="stub"x
2
+class="stub"π
6
=kπ,得x=2kπ-class="stub"π
3

∴f(x)图象的对称中心为(2kπ-class="stub"π
3
,0)
class="stub"x
2
+class="stub"π
6
=kπ+class="stub"π
2
,得x=2kπ+class="stub"2π
3

∴f(x)的对称轴为x=2kπ+class="stub"2π
3

令2kπ-class="stub"π
2
class="stub"x
2
+class="stub"π
6
≤2kπ+class="stub"π
2

得4kπ-class="stub"4
3
π
≤x≤4kπ+class="stub"2
3
π

∴f(x)的递增区间为[4kπ-class="stub"4
3
π
,4kπ+class="stub"2
3
π
]
(II)由x∈[0,π],得class="stub"x
2
+class="stub"π
6
∈[class="stub"π
6
,class="stub"2
3
π]

sin(class="stub"x
2
+class="stub"π
6
)∈[class="stub"1
2
,1]

∴函数f(x)值域为[1,2]

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