f-x-x-m-x-6-and-f-is-strictly-monoton-find-the-value-s-of-m-m-Z-

Question Number 165552 by mnjuly1970 last updated on 03/Feb/22

f (x) = \frac{x + m}{∣ x ∣ + 6} a n d f i s

s t r i c t l y m o n o t o n . f i n d t h e v a l u e (s)

o f m . m \in Z

Answered by TheSupreme last updated on 03/Feb/22

f^{'} (x) > 0

\frac{∣ x ∣ + m - s g n (x) (x + m)}{{(∣ x ∣ + 6)}^{2}} = \frac{∣ x ∣ + m - ∣ x ∣ - s g n (x) m}{{(∣ x ∣ + 6)}^{2}} > 0 \to m (1 - s g n (x)) > 0

∄ m ∣ \forall x \in R : f (x) i s s t r i c t l y m o n o t o n

f (x) i s m o n o t o n f o r m = 0 \to f (x) = s g n (x)

Answered by mahdipoor last updated on 03/Feb/22

\Rightarrow f = {\begin{cases} \frac{x + m}{x + 6} x > 0 \\ \frac{x + m}{6 - x} x ⩽ 0 \end{cases} \Rightarrow f^{^{'}} = {\begin{cases} \frac{6 - m}{{(x + 6)}^{2}} x < 0 \\ \frac{m + 6}{{(6 - x)}^{2}} x ⩾ 0 \end{cases}

{\begin{cases} 6 - m < 0 & m + 6 < 0 \Rightarrow ∄ \\ 6 - m > 0 & m + 6 > 0 \Rightarrow - 6 < m < 6 \end{cases}

m \in Z \Rightarrow m \in {- 5, - 4, \dots, 4, 5}