Show-that-the-function-y-x-5-has-no-derivative-at-x-5-

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

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

Answered by ARUNG_Brandon_MBU last updated on 04/Feb/23

f^{'} (5) = \lim_{x \to 5} \frac{f (x) - f (5)}{x - 5} = \lim_{x \to 5} \frac{∣ x - 5 ∣}{x - 5}

\lim_{x \to 5^{>}} \frac{∣ x - 5 ∣}{x - 5} = \lim_{x \to 5^{>}} \frac{x - 5}{x - 5} = 1

\lim_{x \to 5^{<}} \frac{∣ x - 5 ∣}{x - 5} = \lim_{x \to 5^{<}} \frac{5 - x}{x - 5} = - 1

\lim_{x \to 5^{>}} \frac{∣ x - 5 ∣}{x - 5} \neq \lim_{x \to 5^{<}} \frac{∣ x - 5 ∣}{x - 5}

hence f^{'} (5) does not exist .