Question

f(x+y)=fxfy,且f'0=2，求fx，高数

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Answer 1

f(x+y)=f(x).f(y)
y=0
f(x)= f(0).f(x)
f(0)=1
-------------
f'(0)=2
lim(h->0) [f(h)- f(0)]/h =2
lim(h->0) [f(h+0)- f(0)]/h =2
lim(h->0) f(0) [f(h)- 1]/h =2
=>
h->0
f(h)~ 1+2h
-----------------------
f'(x)
= lim(h->0) [f(x+h) -f(x) ]/h
= lim(h->0) f(x) [f(h) -1) ]/h
=2f(x)
f'(x) = 2f(x)
f(x) = Ae^(2x)
f(0)=1
A=1
ie
f(x) =e^(2x)