## TuringTest Group Title Suppose that $$f$$ satisfies the equation $f(x+y)=f(x)+f(y)+x^2y+xy^2$for all real numbers $$x$$ and $$y$$. Suppose further that$~~~~~~~~~~~~~\lim_{x\to0}\frac{f(x)}x=1$a) Find $$f(0)$$ b) Find $$f'(0)$$ c) Find $$f'(x)$$ one year ago one year ago

1. TuringTest Group Title

I found $$f(0)=0$$ but now I am not seeing how to proceed. Just a hint would be nice.

2. TuringTest Group Title

I further think I proved that the function is odd, though I think that doesn't really help me :/

3. Zarkon Group Title

use the definition of a derivative

4. Zarkon Group Title

the fact that it is odd does seem to help with the last part

5. TuringTest Group Title

I was thinking of using the definition of the derivative somehow but couldn't figure out how to get it into the right form....

6. Zarkon Group Title

$f'(0)=\lim_{x\to0}\frac{f(x)-f(0)}{x-0}$

7. TuringTest Group Title

limit f(x)-f(0) something...

8. TuringTest Group Title

ah dang but... oh, okay I guess I was trying to get things algebraically manipulated there, but I see now that since x=f(x) at x=0 the limit $$is$$ the definition got it :)

9. TuringTest Group Title

okay so that is f'(0)=1, on to the last part you say the oddness could help?

10. Zarkon Group Title

yes

11. TuringTest Group Title

can I use the derivative of an odd function is even?

12. Zarkon Group Title

don't need it

13. Zarkon Group Title

use the definition of a derivative again

14. TuringTest Group Title

that's what I'm doing, let me toy for a sec...

15. Zarkon Group Title

I found f(x) ;)

16. TuringTest Group Title

haha, dang okay then go ahead and give me a little more help on finding f'(x) they don't even ask for f(x), jeez

17. TuringTest Group Title

...if you would be so kind

18. Zarkon Group Title

if you know f'(x) and that f(0)=0 then finding f(x) is really easy :)

19. Zarkon Group Title

$f'(x)=\lim_{y\to x}\frac{f(y)-f(x)}{y-x}$

20. Zarkon Group Title

$f(y)-f(x)=f(y)+f(-x)$

21. Zarkon Group Title

does that help?

22. Zarkon Group Title

ok

23. KingGeorge Group Title

If it helps, it's easier for me to see it when I write the derivative as $f'(x)=\lim_{y\to0} \frac{f(x+y)-f(x)}{y}$

24. TuringTest Group Title

that would be $\lim_{y\to0}{f(y)+x^2y+xy^2\over y}=1+\lim...$ah... it's $$f'(x)=x^2+1$$, right ? :)

25. Zarkon Group Title

yes

26. TuringTest Group Title

sweet, thanks y'all I still want to see how your way was gonna go @Zarkon , but I'll work on that myself for a while

27. Zarkon Group Title

gotta go...fun problem...I'll post it later if you like

28. TuringTest Group Title

always a pleasure :)