设f(x)在[a,b]上连续,在(a,b)内可导,且当x∈(a,b)时,f(x)≠0.若f(a)=f(b)＝0..设f(x)在[a,b]上连续,在(a,b)内可导,且当x∈(a,b)时,f(x)≠0.若f(a)=f(b)＝0.证明：对任意实数k,存在点ε（a＜ε＜b）,使得f‘（ε）/

Question

设f(x)在[a,b]上连续,在(a,b)内可导,且当x∈(a,b)时,f(x)≠0.若f(a)=f(b)＝0..
设f(x)在[a,b]上连续,在(a,b)内可导,且当x∈(a,b)时,f(x)≠0.若f(a)=f(b)＝0.证明：对任意实数k,存在点ε（a＜ε＜b）,使得f‘（ε）/f（ε）＝k.

Answer 1

令F(x)＝e^(－kx)f(x),则F(x)满足闭区间连续,开区间可微,且
F'(x)＝e^(－kx)(f'(x)－kf(x)),F(a)＝F(b)＝0.
由Rolle中值定理,存在c位于(a,b),使得
F'(c)＝0.由于e^(－kc)不等于0,f(c)不等于0,化简
即得结论.