Question and Answers Forum

All Questions      Topic List

Differentiation Questions

Previous in All Question      Next in All Question      

Previous in Differentiation      Next in Differentiation      

Question Number 186445 by myint last updated on 04/Feb/23

Show that the function y = ∣ x −5 ∣ has no derivative at x = 5.

$$\mathrm{Show}\:\:\mathrm{that}\:\:\mathrm{the}\:\:\mathrm{function}\:\:\mathrm{y}\:=\:\:\mid\:\mathrm{x}\:−\mathrm{5}\:\mid\:\:\mathrm{has}\:\:\mathrm{no}\:\:\mathrm{derivative}\:\:\mathrm{at}\:\:\mathrm{x}\:\:=\:\mathrm{5}. \\ $$

Answered by ARUNG_Brandon_MBU last updated on 04/Feb/23

f ′(5)=lim_(x→5) ((f(x)−f(5))/(x−5))=lim_(x→5) ((∣x−5∣)/(x−5)) lim_(x→5^> ) ((∣x−5∣)/(x−5))=lim_(x→5^> ) ((x−5)/(x−5))=1 lim_(x→5^< ) ((∣x−5∣)/(x−5))=lim_(x→5^< ) ((5−x)/(x−5))=−1 lim_(x→5^> ) ((∣x−5∣)/(x−5)) ≠ lim_(x→5^< ) ((∣x−5∣)/(x−5)) hence f ′(5) does not exist.

$${f}\:'\left(\mathrm{5}\right)=\underset{{x}\rightarrow\mathrm{5}} {\mathrm{lim}}\frac{{f}\left({x}\right)−{f}\left(\mathrm{5}\right)}{{x}−\mathrm{5}}=\underset{{x}\rightarrow\mathrm{5}} {\mathrm{lim}}\frac{\mid{x}−\mathrm{5}\mid}{{x}−\mathrm{5}} \\ $$$$\underset{{x}\rightarrow\mathrm{5}^{>} } {\mathrm{lim}}\frac{\mid{x}−\mathrm{5}\mid}{{x}−\mathrm{5}}=\underset{{x}\rightarrow\mathrm{5}^{>} } {\mathrm{lim}}\frac{{x}−\mathrm{5}}{{x}−\mathrm{5}}=\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{5}^{<} } {\mathrm{lim}}\frac{\mid{x}−\mathrm{5}\mid}{{x}−\mathrm{5}}=\underset{{x}\rightarrow\mathrm{5}^{<} } {\mathrm{lim}}\frac{\mathrm{5}−{x}}{{x}−\mathrm{5}}=−\mathrm{1} \\ $$$$\underset{{x}\rightarrow\mathrm{5}^{>} } {\mathrm{lim}}\frac{\mid{x}−\mathrm{5}\mid}{{x}−\mathrm{5}}\:\neq\:\underset{{x}\rightarrow\mathrm{5}^{<} } {\mathrm{lim}}\frac{\mid{x}−\mathrm{5}\mid}{{x}−\mathrm{5}} \\ $$$$\:\mathrm{hence}\:{f}\:'\left(\mathrm{5}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{exist}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com