Determine-all-the-derivable-functions-f-R-R-which-satisfy-f-x-3-f-0-f-x-f-0-x-2-x-0-

Question Number 153162 by mathdanisur last updated on 05/Sep/21

Determine all the derivable functions

f : R \to R which satisfy

\frac{f (x^{3}) - f (0)}{f (x) - f (0)} = x^{2}; \forall x \neq 0

Answered by aleks041103 last updated on 05/Sep/21

l e t g (x) = f (x) - f (0)

g (x^{3}) = x^{2} g (x) \Rightarrow \frac{g (x^{3})}{x^{3}} = \frac{g (x)}{x}

l e t h (x) = g (x) / x = \frac{f (x) - f (0)}{x}

h (x^{3}) = h (x)

b y i n d u c t i o n

h (x) = h (x^{3^{n}}) = h (x^{3^{- n}}), f o r \forall n \in N

l e t 1 > ε > 0 .

c a s e 1 : x \in (0, 1)

w e c a n a l w a y s f i n d s o m e n s u c h t h a t

1 > a = x^{3^{- n}} > 1 - ε

T h e n f o r \forall x \in (0, 1) a n d a n y 1 > ε > 0,

e x i s t s a n u m b e r a \in [1 - ε, 1) s u c h t h a t

h (x) = h (a) .

S i n c e w e c a n t a k e ε t o b e a r b i t r a i r l y

s m a l l, t h e n

h (x) = \lim_{ε \to 0} h (1 > a > 1 - ε)

S i n c e f (x) i s d i f f e r e n t i a b l e, t h e n h (x)

i s t o o (f o r x \neq 0) a n d s o h i s c o n t i n u o u s

\Rightarrow h (x \in (0, 1)) = \lim_{ε \to 0} h (1 > a > 1 - ε) = h (1) .

c a s e 2 : x \in (1, \infty)

b y a n a l o g y, w e c a n a l w a y s f i n d n s u c h t h a t

x^{3^{- n}} \in (1, 1 + ε) .

T h e n a g a i n

h (x > 1) = \lim_{ε \to 0} h (1 < a < 1 + ε) = h (1)

C o n c l u s i o n :

h (x > 0) = h (1) = a = c o n s t .

A n a l o g o u s l y o n e c a n p r o v e

h (x < 0) = h (- 1) = b = c o n s t .

T h e n

f (x) = {\begin{cases} a x + f (0), x ⩾ 0 \\ b x + f (0), x < 0 \end{cases}

B u t f o r f (x) t o b e d i f f . a t x = 0 w e n e e d

a = b .

\Rightarrow f (x) = a x + f (0) i s t h e o n l y s o l u t i o n!

Commented by mathdanisur last updated on 06/Sep/21

Thank you ser nice