Question

一道导数题若函数y=f(x)满足f'(x)>f(x),则当a>0时,f(a)与a^ef(o)之间的大小关系.（a^ef(

Answer 1

设F(x)=f(x)/e^x,则

F'(x)=[f'(x)e^x-f(x)e^x]/e^(2x)=[f'(x)-f(x)]/e^x>0

所以,F(x)单增

所以,F(a)>F(0)

即f(a)/e^a>f(0)

所以,f(a)>e^af(0)

设g(a)=alne-elna=a-elna,则g'(a)=a-e/a

当00,g(a)单增

所以,g(a)>g(e)=e-e=0

所以,alne-elna>0,即e^a>a^e

所以,f(a)>e^af(0)>a^ef(0)