# What do I do when I am trying to find the absolute minimum point of a function and function isn't differentiable at that point?

Consider the function $f(x) = |x|$, obviously this function is not differentiable at $x = 0$. But what does that mean for the absolute minimum point? Do I answer the question with simply "the function isn't differentiable at $x = 0$" or do I state that it isn't differentiable but proceed to write the point anyways?

• You would certainly observe that $f$ has an absolute minimum at $x=0$. You could most easily justify this by appealing to the definition of the absolute value. If you want to bring in calculus, however, you can observe that $f'(x)=-1<0$ for $x<0$, and $f'(x)=1>0$ for $x>0$, so either $f(0)$ is undefined, or the function has an absolute minimum at $x=0$, Since $f(0)$ is defined, the function has an absolute minimum there. – Brian M. Scott Dec 13 '16 at 23:02