# Differentiability of Max Function?

• varygoode
In summary, the conversation discusses determining where a function is differentiable based on its partial derivatives. The first function, f, is differentiable everywhere except at x=0, while the second function, g, is not differentiable at any point where one or more of its variables is equal to zero. The conversation also mentions the function g(x,y)=max(|x|,|y|) as an example of a function that is differentiable at some points but not others.
varygoode

## 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!

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

Dick said:
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.

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

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.

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|).

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?

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.

## 1. What are points of differentiability?

Points of differentiability are points on a function where the derivative exists. In other words, the slope of the function at these points can be defined.

## 2. How do you find points of differentiability?

To find points of differentiability, you need to take the derivative of the function and then solve for values of x where the derivative exists. These values are the points of differentiability.

## 3. Why are points of differentiability important?

Points of differentiability are important because they help us understand the behavior of a function. They can be used to determine the maximum and minimum values of a function, as well as the concavity of the function.

## 4. Are all points on a function differentiable?

No, not all points on a function are differentiable. Points where the derivative does not exist are called points of non-differentiability. This can happen at sharp corners, cusps, or discontinuities in the function.

## 5. Can a function have an infinite number of points of differentiability?

Yes, a function can have an infinite number of points of differentiability as long as the function is continuous and differentiable at every point. An example of this is a polynomial function.

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