Finding a "best" derivative of a function that is not differentiable at a point
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!
Re: Finding a "best" derivative of a function that is not differentiable at a point
In short, are you trying to find derivatives or maxima/minima for functions where derivatives don't exist?
One suggestion I have is if you fit the function to one that is continuous and differentiable by projecting it to say an orthogonal fourier polynomial or orthogonal function basis.
You can find out more about this by looking at fourier analysis.
By choosing the right kinds of functions, you can generate a similar function that preserves derivative information at the turning points.