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In my spare time, I like to play with math problems, sometimes Taylor expands, sometimes integral to find the derivative. I asked myself a math problem for reading:
f(x+y)=f(x)f(y) holds true for any real number x, y, and f'(0)>0, prove:
(1) f(x)>0
(2) f'(x)>0 and f'(x) monotonically increasing
(3) f(x)=e^(f'(0)x)
prove:
(1) f(x)=f(x/2)f(x/2), 故f(x)>=0
Let k exist so that f(k)=0
Then f(x+k)=f(x)f(k)=0
That is f(x+k)=0
That is, f(x)=0, which contradicts f'(0)>0,
So there is no k so that f(k)=0
Therefore f(x)>0 (it can also be proved that f(x) monotonically increases)
(2) f(x)=f(x)f(0), 且f(x)>0,
So f(0)=1
When h->0,
f'(x) = lim(f(x+h)-f(x))/h
= lim(f(x)f(h)-f(x))/h
= f(x)lim(f(h)-1)/h
= f(x)lim(f(h)-f(0))/h
= f(x)f'(0)>0
So f'(x) exists, and f'(x)>0
则f''(x)= f'(x)f'(0)>0
Therefore, f'(x) increases monotonically.
(3) Let g(x)=f(x)e^(-f'(0)x)
Then g'(x)=0
I.e. g(x)=c
That is f(x)e^(-f'(0)x)=c
即 f (x) = ce ^ (f '(0) x)
And f(0)=1, so c=1
So f(x)=e^(f'(0)x)
To put it bluntly, these deterministic things are simple things.
When I was a student, I dealt with certainty every day. Deterministic things make people comfortable, but also easy to be complacent and waste time.
Entering society, after seeing through certainty, take the initiative to abandon certainty, explore and embrace uncertainty.
This is my heart.