Question

The derivative of the function $f\left(x\right)=\sqrt{x^2-2x+1}$ in the interval $[0,2]$ is

• A. $-1$
• B. $1$
• C. $0$
• D. does not exist

Answer 1

Answer: (D)

does not exist

$f\left(x\right)=\sqrt{x^2-2x+1}$, can be written as $f\left(x\right)=|x-1|$
$f\left(x\right)=\{\begin{array}{c}1-x;0\le x<1\\ x-1;1\le x\le 2\end{array}$
$f^{\prime }\left(x\right)=\{\begin{array}{c}-1;0\le x<1\\ 1;1<x\le 2\end{array}$
$\because f^{\prime }\left(1^{-}\right)\ne f^{\prime }\left(1^{+}\right)$
$\therefore$ Derivative of $f\left(x\right)$ does not exist at $x=1$ in interval $[0,2]$.