已知f(x)=sinx+cosx,f′(x)=3f(x),f′(x)为f(x)的导数,则sin2x-3cos2x+1=()A.139B.116C.-149D.-116-数学

题目简介

已知f(x)=sinx+cosx,f′(x)=3f(x),f′(x)为f(x)的导数,则sin2x-3cos2x+1=()A.139B.116C.-149D.-116-数学

题目详情

已知f(x)=sinx+cosx,f′(x)=3f(x),f′(x)为f(x)的导数,则
sin2x-3
cos2x+1
=(  )
A.
13
9
B.
11
6
C.-
14
9
D.-
11
6
题型:单选题难度:中档来源:不详

答案

∵f(x)=sinx+cosx,∴f′(x)=cosx-sinx,
又f′(x)=3f(x)=3sinx+3cosx,∴cosx-sinx=3sinx+3cosx,cosx=-2sinx,tanx=-class="stub"1
2

sin2x-3
cos2x+1
=
sin2x-3(cos2x+sin2x)
cos2 x+( cos2x+sin2x)
=
-2sin2x-3cos2x
2cos2x+sin2x
=
-2tan2x-3
2+tan2x
=
-2×class="stub"1
4
-3
2+class="stub"1
4
=-class="stub"14
9

故选C.

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