已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最值已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最大值和最小值.
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![已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最值已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最大值和最小值.](/uploads/image/z/10532778-42-8.jpg?t=%E5%B7%B2%E7%9F%A5f%28x%29%3Dasinx%2Bbcosx%2C%E5%AE%9A%E7%A7%AF%E5%88%86%EF%BC%880%E5%88%B0%CF%80%2F2%29f%28x%29dx%3D4%2C%E5%AE%9A%E7%A7%AF%E5%88%86%EF%BC%880%E5%88%B0%CF%80%2F6%EF%BC%89f%28x%29dx%3D%287-3%E6%A0%B9%E5%8F%B73%EF%BC%89%2F2%2C%E6%B1%82f%28x%29%E7%9A%84%E6%9C%80%E5%80%BC%E5%B7%B2%E7%9F%A5f%28x%29%3Dasinx%2Bbcosx%2C%E5%AE%9A%E7%A7%AF%E5%88%86%EF%BC%880%E5%88%B0%CF%80%2F2%29f%28x%29dx%3D4%2C%E5%AE%9A%E7%A7%AF%E5%88%86%EF%BC%880%E5%88%B0%CF%80%2F6%EF%BC%89f%28x%29dx%3D%287-3%E6%A0%B9%E5%8F%B73%EF%BC%89%2F2%2C%E6%B1%82f%28x%29%E7%9A%84%E6%9C%80%E5%A4%A7%E5%80%BC%E5%92%8C%E6%9C%80%E5%B0%8F%E5%80%BC.)
已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最值已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最大值和最小值.
已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最值
已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最大值和最小值.
已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最值已知f(x)=asinx+bcosx,定积分(0到π/2)f(x)dx=4,定积分(0到π/6)f(x)dx=(7-3根号3)/2,求f(x)的最大值和最小值.
f(x)=asinx+bcosx
∫【x=0→π/2】f(x)dx=∫【x=0→π/2】(asinx+bcosx)dx
=a∫【x=0→π/2】sinxdx+b∫【x=0→π/2】cosxdx
=a【x=0→π/2】-cosx+b【x=0→π/2】sinxdx
=a[-cos(π/2)+cos0]+b(sin(π/2)-sin0)
=a+b
依题意,有:
a+b=4……………………………………………………(1)
∫【x=0→π/6】f(x)dx=∫【x=0→π/6】(asinx+bcosx)dx
=a∫【x=0→π/6】sinxdx+b∫【x=0→π/6】cosxdx
=a【x=0→π/6】-cosx+b【x=0→π/6】sinxdx
=a[-cos(π/6)+cos0]+b(sin(π/6)-sin0)
=a(2-√3)/2+(b√3)/2
=a+√3(b-a)/2
依题意,有:
a+√3(b-a)/2=(7-3√3)/2……………………………………(2)
由(1)得:a=4-b……………………………………………(3)
代(3)入(2),有:
4-b+√3[b-(4-b)]/2=(7-3√3)/2
4-b+√3(2b-4)/2=(7-3√3)/2
(√3-1)b+4-2√3=7/2-(3/2)√3
(√3-1)b=(√3-1)/2
b=1/2
代入(3),有:a=4-1/2
解得:a=7/2
所以:f(x)=(7/2)sinx+(1/2)cosx
f(x)=(1/2)(7sinx+cosx)
f(x)=[√(7^2+1^2)/2]{[7/√(7^2+1^2)]sinx+[1/√(7^2+1^2)]cosx}
f(x)=[(√50)/2][(7/√50)sinx+(1/√50)cosx]
不妨设:7/√50=cosα,则:1/√50=sinα
代入上式,有:
f(x)=[(√50)/2](cosαsinx+sinαcosx)
f(x)=[(√50)/2]sin(x+α)
f(x)=[(5√2)/2]sin(x+α)
因为:-1≤sin(x+α)≤1
所以:-(5√2)/2≤[(√50)/2]sin(x+α)≤(5√2)/2
即:-(5√2)/2≤f(x)≤(5√2)/2
因此:f(x)的最大值是(5√2)/2,最小值是-(5√2)/2.