Answer to Question #303201 in Real Analysis for Nikhil Singh

Question

The function f: R→R defined by f(x)= | x-1|+ | 3-x| is differentiable at x= 4.

True or false with full explanation

Answer 1

$$f(x)= \begin{cases} -2x+4 &x<1 \\ 2 &1\le x<3 \\ 2x-4 & x\ge 3 \end{cases}$$

For $x\ge 3, f(x)=2x-4$

$$f'(x)=2, x>3$$

Therefore the function $f: R→R$ defined by $f(x)= | x-1|+ | 3-x|$ is differentiable at $x= 4.$