let-f-x-1-x-2-x-1-find-the-domain-and-f-1-2-if-exist-ceiling-function-

Question Number 86894 by M±th+et£s last updated on 01/Apr/20

l e t f (x) = \frac{1 - x^{2}}{⌈ x - 1 ⌉}

f i n d t h e d o m a i n a n d f^{'} (- \frac{1}{2}) i f e x i s t .

⌈ \dots ⌉ c e i l i n g f u n c t i o n

Commented by mathmax by abdo last updated on 01/Apr/20

i f [. .] m e a n i n t e g r p a r t w e h a v e f (x) = \frac{1 - x^{2}}{[x] - 1}

l e t [x] = p \Rightarrow p ⩽ x < p + 1 \Rightarrow f (x) = \frac{1 - x^{2}}{p - 1} a n d f o r p \neq 1 w e g e t

f^{^{'}} (x) = \frac{1}{p - 1} (- 2 x) = \frac{- 2 x}{p - 1}

x = - \frac{1}{2} \Rightarrow [- \frac{1}{2}] = - 1 \Rightarrow f^{^{'}} (- \frac{1}{2}) = \frac{(- 2) \times (- \frac{1}{2})}{- 2} = - \frac{1}{2}

d o m a i n o f f

[x] - 1 = 0 \Leftrightarrow [x] = 1 \Rightarrow 1 ⩽ x < 2 \Rightarrow D_{f} = R - [1, 2 [

Commented by M±th+et£s last updated on 01/Apr/20

⌈ \dots ⌉ s m a l l e s t i n t e g e r f u n c t i o n