设f(x)在[a,b]上连续,在(a,b)内可导且f(a)=f(b)=1.证:存在ζ,η∈(a,b),使e^(η-ζ)[f(η)+f'(η)]=1
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![设f(x)在[a,b]上连续,在(a,b)内可导且f(a)=f(b)=1.证:存在ζ,η∈(a,b),使e^(η-ζ)[f(η)+f'(η)]=1](/uploads/image/z/8506371-3-1.jpg?t=%E8%AE%BEf%28x%29%E5%9C%A8%5Ba%2Cb%5D%E4%B8%8A%E8%BF%9E%E7%BB%AD%2C%E5%9C%A8%28a%2Cb%29%E5%86%85%E5%8F%AF%E5%AF%BC%E4%B8%94f%28a%29%3Df%28b%29%3D1.%E8%AF%81%EF%BC%9A%E5%AD%98%E5%9C%A8%CE%B6%2C%CE%B7%E2%88%88%28a%2Cb%29%2C%E4%BD%BFe%5E%28%CE%B7-%CE%B6%29%5Bf%28%CE%B7%29%2Bf%27%28%CE%B7%29%5D%3D1)
设f(x)在[a,b]上连续,在(a,b)内可导且f(a)=f(b)=1.证:存在ζ,η∈(a,b),使e^(η-ζ)[f(η)+f'(η)]=1
设f(x)在[a,b]上连续,在(a,b)内可导且f(a)=f(b)=1.证:存在ζ,η∈(a,b),使e^(η-ζ)[f(η)+f'(η)]=1
设f(x)在[a,b]上连续,在(a,b)内可导且f(a)=f(b)=1.证:存在ζ,η∈(a,b),使e^(η-ζ)[f(η)+f'(η)]=1
令F(x)=e^xf(x),则F(b)=e^b,F(a)=e^a,F'(x)=e^x(f(x)+f'(x)).
对F(x)用微分中值定理,存在c位于(a,b),使得
(e^b-e^a)/(b-a)=F'(c)=e^c(f(c)+f'(c)).(1)
对函数e^x在[a,b]上用微分中值定理,存在d位于(a,b),使得
(e^b-e^a)/(b-a)=e^d (2)
由(1)和(2)得
e^d=e^c(f(c)+f'(c)),于是
e^(c-d)[f(c)+f(c))]=1,结论成立.
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