Question Number 184893 by mnjuly1970 last updated on 13/Jan/23
$$ \\ $$$$\:\:\:\:{find}\:{the}\:{set}\:{of}\: \\ $$$$\:\:\:\:{critical}\:{points}\:{for}: \\ $$$$\:\:\:\mathrm{1}:\:\:\:{f}\:\left({x}\right)\:=\:\frac{{x}}{\lfloor\:{x}\:\rfloor+\lfloor\:−{x}\rfloor}\:−{x}\:\:\:\: \\ $$$$\:\:\:\:\mathrm{2}\::\:\:{g}\left({x}\right)\:=\:\frac{{x}}{\lfloor\:{x}\:\rfloor\:+\lfloor\:−{x}\:\rfloor}\:+{x} \\ $$
Answered by Mathspace last updated on 13/Jan/23
![1) x in Z ⇒[x]+[−x]=0⇒f is not defined x ∉Z ⇒[x]+[−x]=−1 ⇒ f(x)=(x/(−1))−x=−2x any way we have D_f =R−Z 2)g(x)not defined for x ∈Z and for x∉Z g(x)=(x/(−1))+x=0 ⇒D_g =R−Z](Q184896.png)
$$\left.\mathrm{1}\right)\:{x}\:{in}\:{Z}\:\Rightarrow\left[{x}\right]+\left[−{x}\right]=\mathrm{0}\Rightarrow{f}\:{is}\:{not}\:{defined} \\ $$$${x}\:\notin{Z}\:\Rightarrow\left[{x}\right]+\left[−{x}\right]=−\mathrm{1}\:\Rightarrow \\ $$$${f}\left({x}\right)=\frac{{x}}{−\mathrm{1}}−{x}=−\mathrm{2}{x} \\ $$$${any}\:{way}\:{we}\:{have}\:{D}_{{f}} ={R}−{Z} \\ $$$$\left.\mathrm{2}\right){g}\left({x}\right){not}\:{defined}\:{for}\:{x}\:\in{Z} \\ $$$${and}\:{for}\:{x}\notin{Z}\:\:{g}\left({x}\right)=\frac{{x}}{−\mathrm{1}}+{x}=\mathrm{0} \\ $$$$\Rightarrow{D}_{{g}} ={R}−{Z} \\ $$
Commented by mnjuly1970 last updated on 19/Jan/23
$${thx}\:{alot} \\ $$