Question

高数问题求解 9题

Answer 1

两边同时除以e^(x+y)
得：f(x+y)/e^(x+y)=f(x)/e^x+f(y)/e^y
令：f(x)/e^x=g(x)
上式变为：g(x+y)=g(x)+g(y)
带入x=0，y=0得：g(0)=0
f'(x)=(g(x)+g'(x))e^x
又已知：f'(0)=2
故：g(0)+g'(0)=2
故：g'(0)=2
则：
g'(x)=lim[t→0] (g(t+x)-g(x))/t
=lim[t→0] g(t)/t
=lim[t→0] (g(t)-g(0))/(t-0)
=g'(0)
=2
所以：g(x)=2x+g(0)=2x
f(x)=2xe^x