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设f(x)在[a,b]上连续,在(a,b)内可导且f(

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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