已知函数f(x)可导,且对任何实数x,y满足：f(x+y)=e^xf(y)+e^yf(x)和f'(0)=e 证明：f'(x)=f(x)+e^(x+1)e^x*f(x)
问题描述：
已知函数f(x)可导,且对任何实数x,y满足：f(x+y)=e^xf(y)+e^yf(x)和f'(0)=e 证明：f'(x)=f(x)+e^(x+1)
e^x*f(x)
答
请把e^xf(y)写的清楚点，是相乘还是f(y)作为幂
答
put x=y =0f(0) = f(0) + f(0)=>f(0) = 0f'(x)= lim(y->0)[f(x+y) - f(x)]/y= lim(y->0)[e^xf(y)+e^yf(x) - f(x)]/y= e^x lim(y->0)[f(0+y)-f(0)]/y + f(x) lim(y->0)( e^y - 1)/y= e^xf'(0) + f(x)= e^(x+1) + f(x)