已知函數(shù)f(x)=sinx+cosx,x∈R.
(1)求函數(shù)f(x)的最大值及取得最大值的自變量x的集合;
(2)說明函數(shù)f(x)的圖象可由y=sinx的圖象經(jīng)過的變化得到.
解:(1)∵f(x)=sinx+cosx
=
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sin(x+
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),
∴當(dāng)sin(x+
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)=1,即x=
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+kπ,k∈Z時,函數(shù)取得最大值
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,
此時x的取值集合為{x|x=
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+kπ,k∈Z};
(2)先將y=sinx的圖象向左平移
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個單位,得到y(tǒng)=sin(x+
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)的圖象,
然后再把所得函數(shù)圖象上每個點的縱坐標(biāo)擴大
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倍(橫坐標(biāo)不變)就得到函數(shù)y=f(x)的圖象.
分析:(1)利用輔助角公式將f(x)=sinx+cosx化為f(x)=
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sin(x+
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),利用正弦函數(shù)的性質(zhì)可求得函數(shù)f(x)的最大值及取得最大值的自變量x的集合;
(2)利用三角函數(shù)的圖象變換即可得到答案.
點評:本題考查函數(shù)y=Asin(ωx+φ)的圖象變換,考查正弦函數(shù)的最值,掌握三角函數(shù)的圖象變化與三角函數(shù)的性質(zhì)是解決問題的關(guān)鍵,屬于中檔題.