if-f-x-f-x-a-f-x-a-find-f-x-

Question Number 195971 by mr W last updated on 14/Aug/23

i f f^{'} (x) = \frac{f (x + a) - f (x)}{a}, f i n d f (x) .

Answered by jabarsing last updated on 14/Aug/23

h e l l o m r . w d e a r

f (x) = C (c o n e s t a n t f u n c t i o n) = ?

i s t r u e ?

Commented by mr W last updated on 14/Aug/23

g e n e r a l l y i t c a n b e a n y s t r a i g h t l i n e .

Commented by jabarsing last updated on 14/Aug/23

t h a n k y o u

Answered by MM42 last updated on 14/Aug/23

f (x) = a x + b

Commented by mr W last updated on 14/Aug/23

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Answered by witcher3 last updated on 14/Aug/23

{af}^{'} (x) = f (x + a) - f (x)

\Rightarrow f \in C_{\infty}

by reccurssive \Rightarrow f^{(n)} exist \forall n \in N

f (x + a) - f (x) = \sum_{n ⩾ 0} \frac{f^{n} (x) a^{n}}{n!} = {af}^{'} (x) . . taylor expensin

around x

\Rightarrow \forall n \in N - {0, 1} \Rightarrow f^{(n)} (x) = 0

\Rightarrow f^{'} (x) = c \Rightarrow f (x) = cx + b

ac = c (x + a) + b - cx - b \Rightarrow true

\Rightarrow f (x) = cx + b

Commented by mr W last updated on 14/Aug/23

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