Question

Someone helps me please.
What is the relationship between Ax =0 and Ax=b

Answer 1

I think b = 0

Answer 2

okay

Answer 3

why did u ask

Answer 4

Let r>0 \(f(x)=\begin{cases} x^rsin(1/x) ~~~~if x>0\\0~~~if x\leq 0\end{cases}\)
how to prove f(x) differentiable?

Answer 5

@freckles

Answer 6

Well we need first establish that f is continuous at x=0 and then we need to check out that whole smoothness thing.

Answer 7

\[f(0)=0 \text{ is easy \to see now we just need \to \check if that limit thing is 0 } \\ \lim_{x \rightarrow 0^+} x^r \sin(\frac{1}{x}) =\lim_{x \rightarrow 0^-}0\]
so you know that left limit at x=0 is also obviously 0
we just to make sure the other side is also 0

then we can move on to that smoothness requirement

Answer 8

\[-1 \le \sin(\frac{1}{x}) \le 1 \]
I suggest using squeeze theorem

Answer 9

Still need me, or you got it?

Answer 10

how? when left side is -1 and right side is 1

Answer 11

multiply both sides by x^r

Answer 12

all sides*

Answer 13

I need a lot of help. Appreciate any help

Answer 14

\[-x^r \le x^r \sin(\frac{1}{x}) \le x^r \]

Answer 15

now we know r>0 so you know both -x^r and x^r go to what as x goes to zero?

Answer 16

not quite how squeeze is applied there, but yes, you need to use the periodic nature of sine. Then squeeze if you have already proved it in class

Answer 17

0

Answer 18

by the way the direction of the inequality isn't effected by our multiplication because x>0 since we are only looking at x goes to infinity

so you just did the first part
you showed that the function is continuous at x=0

Answer 19

so for any value of r, we have f'(0) exist, right?

Answer 20

now we need to check the smoothness at x=0

(I thought we were looking at r>0)

Answer 21

ok, that part is ok.
Now another question: :)
for which value of r, f'(x) continuous at 0?

Answer 22

well you know we haven't finished your first problem right?

Answer 23

anyways...for the second question you have here
you need to find f'(x)

Answer 24

do that by finding it in pieces first

Answer 25

like find the derivative of both left and right sides of the function at x=0