f(a+b)=f(a)*f(b) for all a&b suppose f(0)=1&f'(0) exists. Prove:f'(x) f0r all x & that f'(x)=f'(0)*f(x)

Question

This is about Calculus.. Derivatives.. I need it before 24 hours.. Pls help me!! Domo Arigato!

Answer 1

(a + b) = f(a) * f(b)

Recall the definition of a derivative.

f'(x) =
lim [f(x + h) - f(x)]/h
h -> 0

Since f(a + b) = f(a)*f(b), then f(x + h) = f(x)*f(h), so

f'(x) =
lim [f(x)f(h) - f(x)]/h
h -> 0

Factoring out f(x), we get

lim { [f(x) [f(h) - 1] } / h
h -> 0

We can pull the f(x) outside of the limit, giving us

f(x) * lim [f(h) - 1]/h
h -> 0

We know that f(h) = f(0 + h), and 1 = f(0). Let's substitute f(h) and f(1) respectively with those values

f(x) * lim [f(0 + h) - f(0)]/h
h -> 0

But, that is precisely the definition of what f'(0) is (see the definition of the limit, above). Therefore,

f'(x) = f(x) * f'(0)

Answer 2

For the moment assume that F(x) is a polynomial with highest power n. F(F(x)) will contain power 2n and so the equation F(F(x)) = 6x - F(x) cannot be an identity unless n = 1. Let F(x) = ax + b ----> F(F(x)) = a(ax + b) + b = (a^2)x + ab + b F(F(x)) = 6x - F(x) ----> (a^2)x + ab + b = 6x - ax - b (a^2 + a - 6)x + (a + 2)b = 0 For this to be an identity we need a^2 + a - 6 = 0 and (a + 2)b = 0 The first gives a = 2 or -3, the second gives a = -2 or b = 0. y = -3x would not be positive for positive x so a = 2, b = 0 is the only answer. So y = 2x is the only solution IF F(x) is a polynomial. I think that by reference to power series all other functions are eliminated but I am not sure.

Answer 3

Sweetheart, i think your starting equation is invalid... :
if "f (a+b) = f(a)* f(b)" then, it must be "f(a) + f(b)"
- and beacuse it is invalid, how can you use it as analogy for the problem you posted?

Second, the derivative of any constant is Zero?... therefore f'(0) = 0.... therefore in the equation f'(x) = f'(0) * f(x),
f'(x) = 1 ,f'(0) =0 and f(x) is unknown..

let Z = f(x) since it is unknown
therefore 1 = 0*Z
do you think Z exists? I do not think this can be proven.

Answer 4

if f(a+b)= f(a)*f(b) ==> f(x)=Q^x if f(0)=1 ==> Q<>0 so f'(x)=f'(0)*f(x) explicidly. do the math

Answer 5

The only reason for solving or knowing this would be professor,
scientist,nuclear engineer,teacher, or ???????.I am of average intelligence and know nothing about what you have laid out before me.Absolutely useless unless you are going on to be one of the above,this crap is never used by us blue collor regular joes.
Good Luck!

Answer 6

What the????