已知a=(sinx,1),b=(1,cosx)且函数f(x)=ab,f'(x)是f(x)的导函数求函数F(x)=f(x)f'(x)+f^2(x)的最大值和最小正周期
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![已知a=(sinx,1),b=(1,cosx)且函数f(x)=ab,f'(x)是f(x)的导函数求函数F(x)=f(x)f'(x)+f^2(x)的最大值和最小正周期](/uploads/image/z/1699285-13-5.jpg?t=%E5%B7%B2%E7%9F%A5a%3D%28sinx%2C1%29%2Cb%3D%281%2Ccosx%29%E4%B8%94%E5%87%BD%E6%95%B0f%28x%29%3Dab%2Cf%27%28x%29%E6%98%AFf%28x%29%E7%9A%84%E5%AF%BC%E5%87%BD%E6%95%B0%E6%B1%82%E5%87%BD%E6%95%B0F%28x%29%3Df%28x%29f%27%28x%29%2Bf%5E2%28x%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E6%AD%A3%E5%91%A8%E6%9C%9F)
已知a=(sinx,1),b=(1,cosx)且函数f(x)=ab,f'(x)是f(x)的导函数求函数F(x)=f(x)f'(x)+f^2(x)的最大值和最小正周期
已知a=(sinx,1),b=(1,cosx)且函数f(x)=ab,f'(x)是f(x)的导函数
求函数F(x)=f(x)f'(x)+f^2(x)的最大值和最小正周期
已知a=(sinx,1),b=(1,cosx)且函数f(x)=ab,f'(x)是f(x)的导函数求函数F(x)=f(x)f'(x)+f^2(x)的最大值和最小正周期
a = (sinx,1),b = (1,cosx)
ƒ(x) = a • b
= (sinx)(1) + (1)(cosx)
= √2sin(x + π/4)
ƒ'(x) = √2cos(x + π/4)
F(x) = ƒ(x)ƒ'(x) + ƒ²(x)
= [√2sin(x + π/4)][√2cos(x + π/4)] + [√2sin(x + π/4)]²
= sin[2(x + π/4)] + 1 - cos[2(x + π/4)]
= sin(2x + π/4) - cos(2x + π/2) + 1
= cos(2x) - [- sin(2x)] + 1
= sin(2x) + cos(2x) + 1
= √2sin(2x + π/4) + 1
最小正周期T = (2π)/2 = π
最大值 = 1 + √2
最小值 = 1 - √2
f(x)=sinx+cosx
f'(x)=cosx-sinx
F(x)=f(x)f'(x)+f²(x)
=(sinx+cosx)(sinx-cosx)+(sinx+cosx)²
=sin²x-cos²x+1+2sinxcosx
=-cos2x+sin2x+1
=-...
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f(x)=sinx+cosx
f'(x)=cosx-sinx
F(x)=f(x)f'(x)+f²(x)
=(sinx+cosx)(sinx-cosx)+(sinx+cosx)²
=sin²x-cos²x+1+2sinxcosx
=-cos2x+sin2x+1
=-√2(√2/2sin2x-√2/2cos2x)+1
=-√2sin(2x-π/4)+1
F(x)的最大值=√2+1 (当sin(2x-π/4)=-1
最小正周期:2π/2=π
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