已知函数f(x)=sinx+cosx,F(x)=f'(x)[f(x)+f'(x)]-1,f'(x)是f(x)的导函数.(1)若tanx=1/3,求F(X)的值(2)求F(x)的单调减区间
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已知函数f(x)=sinx+cosx,F(x)=f'(x)[f(x)+f'(x)]-1,f'(x)是f(x)的导函数.(1)若tanx=1/3,求F(X)的值(2)求F(x)的单调减区间
已知函数f(x)=sinx+cosx,F(x)=f'(x)[f(x)+f'(x)]-1,f'(x)是f(x)的导函数.(1)若tanx=1/3,求F(X)的值
(2)求F(x)的单调减区间
已知函数f(x)=sinx+cosx,F(x)=f'(x)[f(x)+f'(x)]-1,f'(x)是f(x)的导函数.(1)若tanx=1/3,求F(X)的值(2)求F(x)的单调减区间
∵f(x)=sinx+cosx,
∴f'(x)=cosx-sinx,
∴F(x)=f'(x)[f(x)+f'(x)]-1
=(cosx-sinx)(sinx+cosx+cosx-sinx)-1
=2cos^2x-sin2x-1
=1+cos2x-sin2x-1
= 根号2 cos(2x+π/4 ),
∵tanx=1/3 ,
∴cos2x=cos^2x-sin^2x=(cos^2x-sin^2x)/(cos^2x+sin^2x)=(1-tan^2x)/(1+tan^2x)=4/5 ,
同理可求sin2x=2tanx/(1+tan^2x)=3/5 ,
∴F(x)=根号2 cos(2x+π/4 )=根号2(cos2xcosπ/4 -sin2xsinπ/4 )=1/5 .
∵F(x)=根号2 cos(2x+π/4 ),
∴由2kπ≤2x+π/4 ≤2kπ+π得:kπ-π/8 ≤x≤kπ+3π/8 (k∈Z),
即:x∈[kπ-π/8 ,kπ+3π/8 ](k∈Z),
此即F(x)的单调减区间.
楼上正解
F(x)=(cosx-sinx)[sinx+cosx+cosx-sinx]-1
=2(cosx)^2-sin2x-1
=2(cosx)^2 (1-tanx)-1,
(1)tanx=1/3,
(cosx)^2=1/(1+(tanx)^2)=9/10,
F(X)=2* (9/10 ) *(2/3)-1
=1/5.
(2)
对F(x)求导得-2sin2x-2cos2x,令其<=0得sin2x+cos2x<=0,以后你自己可以算吧
(1)f'(x)=cosx-sinx
F(x)=(cosx-sinx)(sinx+cosx+cosx-sinx)-1
=2(cosx)^2-1-2sinxcosx
=cos2x-sin2x
则F(x)=(2(cosx)^2-2sinxcosx)/((sinx)^2+(cosx)^2)-1
=2(((tanx)^2-2tanx)/((tanx)^2)+1)-1...
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(1)f'(x)=cosx-sinx
F(x)=(cosx-sinx)(sinx+cosx+cosx-sinx)-1
=2(cosx)^2-1-2sinxcosx
=cos2x-sin2x
则F(x)=(2(cosx)^2-2sinxcosx)/((sinx)^2+(cosx)^2)-1
=2(((tanx)^2-2tanx)/((tanx)^2)+1)-1
带入得F(x)=-2
(2)
F(x)=cos2x-sin2x=根号2cos(2x+PI/4)<=0 (PI就是圆周率)
易得
[PI/8+kPI,5PI/8+kPI]
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f'(x)=cosx-sinx
F(x)=(cosx-sinx)(sinx+cosx+cosx-sinx)-1
=2cosx(cosx-sinx)-1
=2cos²x-1-2sinxcosx
=cos2x-sin2x
=√2sin(π/4-2x)
(1)tanx=1/3 ∴sinx/cos...
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f'(x)=cosx-sinx
F(x)=(cosx-sinx)(sinx+cosx+cosx-sinx)-1
=2cosx(cosx-sinx)-1
=2cos²x-1-2sinxcosx
=cos2x-sin2x
=√2sin(π/4-2x)
(1)tanx=1/3 ∴sinx/cosx=1/3 ∴cosx=3sinx
又∵sin²x+cos²x=1 ∴sin²x+9sin²x=1 sin²x=1/10
则f(x)=4sinx f'(x)=2sinx
F(x)=2sinx(4sinx+2sinx)-1=12sin²x-1=1/5
(2)y=sinx的单调递减区间为[π/2+2kπ,3π/2+2kπ]
则F(x)=√2sin(π/4-2x)的单调递减区间为π/2+2kπ≤π/4-2x≤3π/2+2kπ
[-5π/8+kπ,-π/8+kπ]
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