Question

已知函數(shù)f(x)=2acos²x+bsinxcosx,且f(0)=2,f()=+.

(1)求a,b的值;

(2)求函數(shù)f(x)的最大值及取得最大值時(shí)的x的集合;

(3)寫(xiě)出函數(shù)f(x)在［0,π］上的遞減區(qū)間.

Answer 1

解:(1)將條件代入,解方程組得

(2)由(1)有f(x)=2cos²x+2sinxcosx=1+cos2x+sin2x=1+sin(2x+),

∴當(dāng)2x+=2kπ+,即x=kπ+(k∈Z)時(shí),f(x)取得最大值1+2.

(3)∵x∈［0,π］,∴2x+∈［,］,要使函數(shù)f(x)遞減,則應(yīng)2x+∈［,］.

可得x∈［,］,∴函數(shù)f(x)在［0,π］上的遞減區(qū)間為［,］.