One sided differentiation prove

[transgalactic]

i am given two differentiable function f and g .
prove that for u(x)=max(f(x),g(x))
and v(x)=min(f(x),g(x))

there is one sided derivatives
??how to put mim ,max functions into the formula of derivative formula??

f'(x)_+ = \lim_{h \to 0^+} \frac {f(x + h) - f(x)}h (one-sided derivative...from the right)
<br /> f&#039;(x)_- = \lim_{h \to 0^-} \frac {f(x + h) - f(x)}h<br />
(one-sided derivative... from the left)

 

[CompuChip]

So let's fix a point x.
I think you can show that if one function is larger at x then there is some one-sided interval for which it stays the larger one.
So for example, if u(x) = f(x)) then there exists an \epsilon &gt; 0 such that u(y) = f(y) for all y \in [x, x + \epsilon). So then u&#039;(x)_+ = f&#039;(x).

 

[transgalactic]

how to prove it?

 

[CompuChip]

I am assuming you have done / are doing some analysis.
What you could do, for example, is define h(x) := f(x) - g(x).
Then if u(x) := max(f(x), g(x)) is equal to f(x) in some point x₀, that is equivalent to h(x) being greater or equal to 0.

Do you then see how your question reduces from
"Show that there is a right one-sided derivative of u(x)"
to
"Show that if h(x_0) \ge 0 then there is some \epsilon &gt; 0 such that h(x) \ge 0, \, \forall x \in [x_0, x_0 + \epsilon)"

 

[transgalactic]

i understand you explanation
but i can't deside if it the needed proove

deffentiable function is a function which on soome point its
left side derivative equals its write side derivative.

so
if both functions are derivatable.
then it doent matter what function we take

there for sure will be a function where
it has a point for which there is a derivative on both sides
even though we need only one.
so its an over kill
am i correct?

 

[quasar987]

There is something I would like to point out. It does not seem exactly true that if h(x_0) \ge 0 then there is some \epsilon &gt; 0 such that h(x) \ge 0, \, \forall x \in [x_0, x_0 + \epsilon).

For instance, consider f(x)=-|x| and g(x)=-x². Then u(0)=f(0) but in no one sided interval of 0 do we have h(x) \geq 0.

 

[Hurkyl]

For instance, consider f(x)=-|x|

That f isn't differentiable...

 

[quasar987]

Good point, but the idea still stands. Take instead f(x)=-x² and g(x)=-x^4.

 

[CompuChip]

For f(x) = -x² and g(x) = -x⁴, the difference h(x) := f(x) - g(x) actually has the same sign throughout the entire interval ]-1, 1[.

OK, so maybe you have to redefine h(x) as g(x) - f(x) to get it positive; that's irrelevant for the proof.

 

[CompuChip]

i understand you explanation
but i can't deside if it the needed proove

deffentiable function is a function which on soome point its
left side derivative equals its write side derivative.

so
if both functions are derivatable.
then it doent matter what function we take

there for sure will be a function where
it has a point for which there is a derivative on both sides
even though we need only one.
so its an over kill
am i correct?

Note that max(.., ..) is not in itself a differentiable function. For example,
x \mapsto \max(x, -x)
is not differentiable at zero, because it's basically just x \mapsto |x| - even though x \mapsto x, x \mapsto -x are smooth.

So if you plug two differentiable functions into max(.., ..) you don't necessarily get something differentiable. The point of the exercise is, that a one-sided derivative always exists though (in this case, u&#039;(x)_\pm = \pm 1).

 

[transgalactic]

on what basis i can say that max has always one-sided derivative.

in your example you had a formula
<br /> u&#039;(x)_\pm = \pm 1<br />

but in my question its totally abstract

 

[transgalactic]

this is the only equations i can construct

<br /> \mathop {\mathop {\lim }\limits_{x \to x_0 - } \frac{{f(x_{} ) - f(x_0 )}}{{x - x_0 }} = \mathop {\lim }\limits_{x \to x_0 + } \frac{{f(x_{} ) - f(x_0 )}}{{x - x_0 }}}\limits_{} \\<br />
<br /> \mathop {\mathop {\lim }\limits_{x \to x_0 - } \frac{{g(x_{} ) - g(x_0 )}}{{x - x_0 }} = \mathop {\lim }\limits_{x \to x_0 + } \frac{{g(x_{} ) - g(x_0 )}}{{x - x_0 }}}\limits_{} \\ <br />

 

[transgalactic]

"if u(x) = f(x)) then there exists an \epsilon &gt; 0 such that u(y) = f(y) for all y \in [x, x + \epsilon). So then u&#039;(x)_+ = f&#039;(x)."

i don't know how to apply this this to my question

i showed the differentiation equations
i don't know how to continue.

 

[CompuChip]

So I think you are missing the geometric picture that I have in mind and therefore cannot make sense of my algebra either

Start by drawing two differentiable curves, which may or may not intersect in some point. Pick a point x₀ on the x-axis. If the curves don't intersect there, then one will be bigger than the other one in x₀ and a small interval around it. So if you consider the maximum of the two, u = max(f(x), g(x)), then u will just be equal to either f(x) or g(x) in some interval around x₀, so the derivative of u will just be either f'(x) or g'(x), respectively. Since f and g are both differentiable, you actually have a ("two-sided") derivative.

However, it is possible that the curves intersect in x₀, and that u(x) changes from being equal to f(x) to being equal to g(x) - again: draw a picture! Then the derivative of u(x) will be equal to f'(x) on one side, and to g'(x) on the other side. Since f'(x₀) isn't necessarily equal to g'(x₀), you cannot really say that the ("two-sided") limit
\lim_{x \to x_0} \frac{u(x) - u(x_0)}{|x - x_0|}
exists. However, the claim in your exercise is, that if you only take the limit from one side, that is, consider
\lim_{x \downarrow x_0} \frac{u(x) - u(x_0)}{|x - x_0|}
or
\lim_{x \uparrow x_0} \frac{u(x) - u(x_0)}{|x - x_0|}
then the limit does exist (and from the above story I hope you see that it will be equal to the derivative of the function along which you are approaching x₀).

Does the question and picture make sense to you now? I think it's important that you understand why it is true, before you try to show how it is true.

 

[transgalactic]

so you basically say
that maximum will pick either way some function
and in every case around that x_0 it has to be differentiable because its defind
to be differentiable on every x.

correct?

 

[CompuChip]

so you basically say
that maximum will pick either way some function

Yes

and in every case around that x_0 it has to be differentiable because its defind
to be differentiable on every x.

Almost, except that it can happen that it picks different functions on different sides of x_0, so you can't take the limit on both sides as in ordinary derivative... but on each side separately it is defined to be equal to a differentiable function.

 

[transgalactic]

so u(x) and v(x) around x_0 have to pick different functions
because if one is bigger then the other is smaller.

so if f(x)=u(x) then g(x)=v(x) and once again
because f(x) and g(x) are differentiable then we have one sided derivative.

i don't know how to write it thematically but here is a try

<br /> \forall x_0 \in R<br />
<br /> {\rm{ }}\exists \delta {\rm{ &gt; 0 |x - x}}_0 | &lt; \delta {\rm{ }}\mathop {\lim }\limits_{x \to x_0 } u(x) = \mathop {\lim }\limits_{x \to x_0 } v(x)<br />

 

[transgalactic]

how to prove it??