# Functions ?

1. Apr 1, 2009

### ARYT

functions ???

1. The problem statement, all variables and given/known data

I apologize for the title, I don't know what to call these kind of problems.

Q3. Suppose f is function that satisfies the equation

equation 1

For all numbers x and y, suppose also that

equation 2

Find: (i). f(0), (ii). f’(0), (iii). f’(x)

2. Relevant equations

?

3. The attempt at a solution

I have no idea. So what?

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2. Apr 1, 2009

### lanedance

Re: functions ???

hi Aryt

pictures take a while to get viewable.... so can't see yet, can you enter the problem otherwise?

3. Apr 1, 2009

### ARYT

Re: functions ???

equation 1: f(x+y)=f(x)+f(y)+x^2 y+xy^2

equation 2: lim f(x)/x =1 while x tends to zero

4. Apr 1, 2009

### lanedance

Re: functions ???

is that all the info...? if so, then how about considering f(1+0)?

then think about a derivative from first limits at zero...

anyway, see what you think ;)

5. Apr 1, 2009

### ARYT

Re: functions ???

I think we should write sth like: f(-1+1) = f(0) = f(1)- f(-1)+(1)^2 (-1) + (1)(-1^2)
which will be f(1)-f(-1). and we know that when x tends to 0, f(x)/x = 1
so f(1)/1 = 1 and also f(-1)/-1 = 1
Therefore f(0) = 1-1=0 or I dunno, maybe 1+1=2 .

Is it a possible answer for (i)?

ahhhhhh, what kind of question is this? :(

6. Apr 1, 2009

### FedEx

Re: functions ???

I think that it is rather a good question.
Get to the defintion of f'(x).

f'(x)= [f(x+h) - f(h)}/h . Where h tends towards zero. Now think what can you do to f(x+h). Once done with that. You wil get to a dead end. But you can get rid of that by knowing the value of f(0). The value of f(0) cab be found out by replacing x and y in the original equation by 0 and 0. Once done with that you will get the value of F(0).

7. Apr 1, 2009

### ARYT

Re: functions ???

well, if we go for 0 and 0, we will have: f(0) = f(0)+f(0) that is f(0)=2f(0)
and 1=2 ! Great!

8. Apr 1, 2009

### FedEx

Re: functions ???

Well its obvious isnt it. You are divinding by zero. Hence you get ridiculous statements like 1=2. So f(0)=0

9. Apr 1, 2009

### ARYT

Re: functions ???

so f'(0) is also zero?

and f'(x)=1?

I think I got it. Thanks

10. Apr 1, 2009

### lanedance

Re: functions ???

i think you're there, but here's what I meant

$$f(x+y)=f(x)+f(y)+x^2 y+xy^2$$
$$f(0+1)=f(0)+f(1)+0^2.1+0.1^2$$
$$f(1)=f(0)+f(1)$$
$$0=f(0)$$

11. Apr 1, 2009

### FedEx

Re: functions ???

I haven't calculated the value of f'(x). Even if it is equal to 1 how can the value of f'(0) be equal to zero considering that f'(x)=1

12. Apr 1, 2009

### lanedance

Re: functions ???

how did you get f'(0) = 0, is that a guess?
calculate f'(0) first as Fedex implies (use the limit information and the first principles defintino of a derivative)

Last edited: Apr 2, 2009
13. Apr 2, 2009

### ARYT

Re: functions ???

f'(x)= [f(x+h) - f(h)]/h while h tends to zero.

so f'(0)=[f(0+0)-f(0)]/0
f'(0)= 0/0

???

14. Apr 2, 2009

### lanedance

Re: functions ???

remember its not h=0, its the limit as h goes to zero

15. Apr 2, 2009

### ARYT

Re: functions ???

We're under heavy fire C.O., Clarification is needed. :D

lol

Isn't it a bit stupid to say:
We have: f(0) = 0 and Equation 1:

we have lim [f(0+h)-f(h)]/h while h maps to 0.
Thus (h tends to 0): lim f(h)/h - lim f(h)/h = 1-1 = 0

16. Apr 2, 2009

### ARYT

Re: functions ???

I think I've found the problem. You've cited the equation wrongly. It should be:

[f(x+h)-f(x)]/h while h tends to 0. According to equation 1, we have:

[f(x)+f(h)+x^2 h + xh^2 - f(x)] / h => lim f(h)/h + lim (x^2+xh) while h maps to zero.

So f'(x)=1+x^2

and accordign to this we have: f'(0)=1+0=1

Right?

17. Apr 2, 2009

### FedEx

Re: functions ???

Buffon

18. Apr 2, 2009

### FedEx

Re: functions ???

Genuis

PS Sorry for "buffon" I just wanted to bring a dramatic end.

19. Apr 2, 2009

### ARYT

Re: functions ???

No, It's OK. Thanks for the tips. :D

A better dramatic end: lol

Although they're killing my innovation here. For one assignment (C++ programming), I've used only one line to compile a rather complicated program without sth useless (i.e. End of File function: eof.). And the tutor told me very directly, “why are you trying to come up with your own ways”? I gave you a kind of lecture for this question. When you can't solve sth in my way, why you do it in your way?
ahhhh, it's innovation, if you add only one space to the input file, your program will fail, but mine will remain ok, I said! :)

Last edited: Apr 2, 2009