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Question

Let f(x)=cosx(sinx+sin2x+sin2θ),θ is a given const, then max of f(x)is


A

1+csc2θ

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B

1+sin2θ

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C

|cosθ|

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D

|sinθ|

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Solution

The correct option is B

1+sin2θ


f(x)=cosx(sinx+sin2x+sin2θ)f(x)=1secx(sinx+sin2x+sin2θ)secxf(x)=sinx+sin2+sin2θ(secxf(x)sinx)2=sin2x+sin2θ=sec2xf2(x)+sin2x2sinxsecxf(x)=sin2x+sin2θ=f2(x)[1+tan2x]2f(x)tanx=sin2θ=f2(x)tan2x2f(x)tanx+f2(x)sin2θ=04f2(x)4f2(x)(f2(x)sin2θ)=f2(x)1+sin2θ|f(x)|1+sin2θ


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