函数f,g在[a,b]连续,(a,b)可导,f(a)=f(b)=0,证明存在c∈(a,b)使得f'(
问题描述：
函数f,g在[a,b]连续,(a,b)可导,f(a)=f(b)=0,证明存在c∈(a,b)使得f'(
c)+f(c)g'(c)=0
答
考虑h(x)=f(x)e^(g(x)),有h(x)在[a,b]连续,(a,b)可导,且h(a)=h(b)=0.
由罗尔中值定理,存在c∈(a,b)使h'(c)=0.
而h'(c)=(f'(c)+f(c)g'(c))e^(g(c)),其中e^(g(c))不等于0.
故f'(c)+f(c)g'(c)=0.