Question

If f(x) = |x| + |x - 1|, write the value of $\frac{d}{dx}\left(f\right(x\left)\right)$.

Answer 1

$f\left(x\right)=|x|+|x-1|$

$\text{when}x>1$

$f\left(x\right)=x+x-1=2x-1$

$\frac{d}{dx}\left(f\right(x\left)\right)=2$

$\text{when}0<x<1$

$f\left(x\right)=x-x+1=1$

$\frac{d}{dx}\left(f\right(x\left)\right)=0$

$\text{when}x<0$

$f\left(x\right)=-x-x+1=-2x+1$

$\frac{d}{dx}\left(f\right(x\left)\right)=-2$

$\frac{d}{dx}\left(f\right(x\left)\right)=\left\{\begin{array}{c}2,x>1\\ 0,0<x<1\\ -2,x<0\end{array}\right\}$