Differentiability of Max Function?

[varygoode]

Homework Statement

Homework Equations

The Attempt at a Solution

I think to determine where it's differentiable it has something to do with partial derivatives. But I am just so completely clueless on how to even start this guy off that any tips or minor suggestions on where to even begin would be great. Thanks!

 

[Dick]

Start with the first one, f. f is just a sum of absolute values. Where is |x| differentiable for x a real variable??

 

[varygoode]

Start with the first one, f. f is just a sum of absolute values. Where is |x| differentiable for x a real variable??

Everywhere except at x=0.

 

[Dick]

Ok, so if ALL of the x_i's are non-zero, then f is differentiable, right? Where is it not differentiable?

 

[varygoode]

When they're ALL zero? Because if one x_i is non-zero, but the rest are, it's still differentiable?

I don't know why or how to prove it though, and that's what I need help with.

 

[Dick]

If any x_i is equal to zero then the derivative doesn't exist. The ith partial derivative doesn't exist. g is harder, I'm still trying to figure out a good way to describe the set. Try warming up with just two variables. g(x,y)=max(|x|,|y|).

 

[varygoode]

Alright, I think I understand what's going on with f.

So then with g, couldn't I try a similar idea since it is also absolute value? Or no?

 

[Dick]

It's similar but not the same. Like I say, think about g(x,y)=max(|x|,|y|). That IS NOT differentiable at x=3, y=3. It IS differentiable at x=3, y=0. Once you've figured out why, try and figure out a way to describe all of the points where it is not differentiable.