I have a topic I am suppose to investigate and write a small paper about. The question deals with functions that are non-differentiable at one or more points, but "looks like" it "could" have a horizontal tangent at one or more of these points if it could. For example, y = |x| is clearly not differentiable at x = 0 and thus has no horizontal tangent line, but the most natural tangent line there seems to be at x=0, which reveals a relative minima at (0,0). So, the question is what do you do/set equal to zero with these kinds of functions i.e. is there some "derivative-like" quantity/formula analogous to the regular definition of the derivative that would produce these values (x values that appear to have a "best" tangent line despite not being differentiable there). I would then need go into detail about generalzing all the regular theorems such as product, quotient, chain, IVT, etc., but first I need some basic direction on where to start.
Thanks for any advice!