The local minimum value of $f(x)=3+|x|$ on $\mathbf{R}$ is

1. $f$ is not differentiable at $x=0$, so the second derivative test fails. 2. Use the first derivative test: for $x<0$, $f'(x)=-1<0$; for $x>0$, $f'(x)=1>0$. 3. The sign changes $-$ to $+$, so $x=0$ is a local minima. 4. Local minimum value $f(0)=3+0=3$. _Source: syllabus Mathematics Ch 6 "Application of Derivatives", p.19_