Question

State true or false the function f defined by f (x)=| x − 2| is differentiable in [0 ,1 ].

Answer 1

The function is differentiable in [0 ,1] when it has derivative in every point in [0 ,1].

$f'(x)=(|x-2|)'$ can be rewritten as

$f'(x)=(\sqrt{(x-2)^2})'=\frac{((x-2)^2)'}{2\sqrt{(x-2)^2}}=$

$=\frac{2(x-2)}{2\sqrt{(x-2)^2}}=\frac{x-2}{|x-2|}$ , where $x-2\neq0$. So the function $f(x)=|x-2|$ hasthe derivative in every point except $x=2$ .Answer: true, the function $f(x)=|x-2|$ is differentiable in [0 ,1].