Determine-all-functions-f-0-1-such-that-x-0-1-f-x-f-x-f-0-f-1-

Question Number 88003 by Ar Brandon last updated on 07/Apr/20

D e t e r m i n e a l l f u n c t i o n s f [0, 1] \to Ω

s u c h t h a t \forall x \in [0, 1] f^{'} (x) + f (x) = f (0) + f (1)

Commented by mr W last updated on 07/Apr/20

f (0) + f (1) = k

f^{'} (x) + f (x) = k

\frac{d y}{y - k} = - d x

\ln ∣ y - k ∣= - x + C

y = k + {c e}^{- x}

f (0) = k + c

f (1) = k + \frac{c}{e}

f (0) + f (1) = k + c + k + \frac{c}{e} = k

k = - c (1 + \frac{1}{e})

\Rightarrow y = f (x) = c (e^{- x} - 1 - \frac{1}{e})

Commented by Ar Brandon last updated on 07/Apr/20

T h a n k s