设f(x)在[a,b]上连续,在(a,b)内可导,且f(a)=f(b)=0.证明：在（a,b）内至少存在一点c,使f'(c)+df(c)=0这d为任意实数.

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设f(x)在[a,b]上连续,在(a,b)内可导,且f(a)=f(b)=0.证明：在（a,b）内至少存在一点c,使f'(c)+df(c)=0这
d为任意实数.

设g(x)=f(x)e^(dx),
由题意得g(x)在(a,b)上可导,[a,b]内连续,
又g(a)=f(a)e^(da)=0
g(b)=f(b)e^(da)=0
即g(a)=g(b)
对g(x)在[a,b]区间应用罗尔定理,
至少存在一点c,使得
g'(c)=0
即f'(c)e^(dc)+df(c)e^(dc)=0
对上式左右除以e^(dc)可得
原命题得证.

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