## 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 $$f'(x)$$ 2 years ago 2 years ago

1. TuringTest

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

2. TuringTest

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

3. Zarkon

use the definition of a derivative

4. Zarkon

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

5. TuringTest

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

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

7. TuringTest

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

8. TuringTest

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

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

10. Zarkon

yes

11. TuringTest

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

12. Zarkon

don't need it

13. Zarkon

use the definition of a derivative again

14. TuringTest

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

15. Zarkon

I found f(x) ;)

16. TuringTest

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

...if you would be so kind

18. Zarkon

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

19. Zarkon

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

20. Zarkon

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

21. Zarkon

does that help?

22. Zarkon

ok

23. KingGeorge

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

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

yes

26. TuringTest

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

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

28. TuringTest

always a pleasure :)