# 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

## 1 Answer

The extrema of functions are found where the derivative is equal to zero OR the derivative is undefined (although those don't necessarily imply extrema exists at those points). This is the second case.

This is pretty clear simply by looking at the graph of the function. To the left and right of (0,0), the function has higher y-values, and also no longer changes slope, as we can see by looking at the derivative. Therefore, the minimum must be (0,0).