Help with 3 Math Questions: Proving Differentiability & More

[antibody]

1. Suppose that f(a)=g(a) and the left-hand derivative of f at a equals the right-hand derivative of g at a. Define h(x)=f(x) for x<=a, and h(x)=g(x) for x>=a. Prove that h is differentiable at a.

2. Let 0<B<1. Prove that if f satisfies /f(x)/ >= /x/^B and f(0)=0, then f is not differentiable at 0.

the sign / / is absolute value.

3. Let f(x)=x^n for x>=0 and let f(x)=0 for x<=0. Prove that f^(n-1)exists and find a formula for it, but that f^(n) (0) does not exist.


Can someone help me out with these problems? thanks a lot!

 

[cristo]

You need to show some work on homework questions before we can help you.

 

[quantumdude]

You need to show some work on homework questions before we can help you.

Yep. That, and homework goes in the Homework Help section, not the Math section.

So, antibody, let's see what are you got so far.

 

[antibody]

You need to show some work on homework questions before we can help you.

for the second one , i know how to prove the converse, like let B>1, if f satisfies /f(x)/ <=/x/^B, prove that f is differentiable at 0,

this one will be easier, first let x=0 then f(0)=0, and i know to prove some fn is differentiable at some point x, it means to prove lim(h->0) f(x+h)-f(x) /h = some number( here is 0 since the prob has given)

i guess i can do the same thing to the second one, but i m not sure how to write a religious proof.

 

[antibody]

and same thing happens to quesntion no.3 ...

if the question gives me some precise function, i probably can solve it,
but this one i am still working on it, my idea is
x^n when x>=0
f(x)=
0 when x<=0

so the f ' (x) = n x^(n-1) when x>0 and f ' (x)=0 when x<0 then use the left and right limit to see if f ' (x) exists when x=0. Right?

and again..i am not very familiar with this kind of proof right now,, since we just spent a week on this topic, and i am not sure when we need to use delta-epsilon proof on the question...