Finding f(0) for the following function

[KataKoniK]

Hi,

Can anyone here help me do the following question? I tried isolating f(x) and f(y), but it doesn't really seem to go anywhere.

Q: Suppose f is a functionb which satisfies

f(x+y) = f(x) + f(y) +xy - x^3y + xy^3 - y^4 for all x, y, are elements of real numbers. Suppose, furthermore, that

lim f(x) / x = 1
x -> 0

a) Find f(0)
b) Show that f is differentiable at 0 (any tips)?

 

[Muzza]

a) What happens if you set x = y? How can this be used to find f(0)?
b) Yes, simply use f'(a) = lim(x->a) (f(x) - f(a)) / (x - a). After you've found f(0), this expression can be simplified in a nice way.

 

[matt grime]

I think the fact that for all x, x+0=x would probably help.

 

[HallsofIvy]

Actually, just saying that $lim_{x\rightarrow 0} \frac{f(x)}{x}$ exists tells you what f(0) is!

 

[shmoe]

Actually, just saying that $lim_{x\rightarrow 0} \frac{f(x)}{x}$ exists tells you what f(0) is!

Asuming you've already shown (or were given) that f is continuous as 0.

To find f(0) you can also use the fact that 0+0=0.

 

[trap]

can someone please actually post the answer to this question?...i tried to do this, but i still don't exactly how to do this question with the hints provided, thanks

 

[Galileo]

You wish to find f(0), so just fill in x=y=0 and see what you get.

f(x+y) = f(x) + f(y) +xy - x^3y + xy^3 - y^4
for x=y=0 becomes
f(0)=2f(0)

what does this say about f(0)?

 

[trap]

oh, i see, i get it now, thanks
and for part b), I'm still having trouble understanding, and you provide an answer to this too, thanks alot

 

[Galileo]

Write down the definition of the derivative at x=0.

 

[KataKoniK]

Thanks everyone.

How would I find if the function is differntiable for all x and then find f'(x)?

 

[Galileo]

No seriously: Write down the definition of the derivative at x=0.
The answer should pop in your face.

'Definition': The derivative of a function f at a point x is:
$$\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$$