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Finding f(0) for the following function

  1. Nov 7, 2004 #1
    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)?
     
  2. jcsd
  3. Nov 7, 2004 #2
    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.
     
  4. Nov 8, 2004 #3

    matt grime

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    I think the fact that for all x, x+0=x would probably help.
     
  5. Nov 9, 2004 #4

    HallsofIvy

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    Actually, just saying that [itex]lim_{x\rightarrow 0} \frac{f(x)}{x}[/itex] exists tells you what f(0) is!
     
  6. Nov 9, 2004 #5

    shmoe

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    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.
     
  7. Nov 10, 2004 #6
    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
     
  8. Nov 10, 2004 #7

    Galileo

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    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)?
     
  9. Nov 10, 2004 #8
    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
     
  10. Nov 10, 2004 #9

    Galileo

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    Write down the definition of the derivative at x=0.
     
  11. Nov 10, 2004 #10
    Thanks everyone.

    How would I find if the function is differntiable for all x and then find f'(x)?
     
  12. Nov 10, 2004 #11

    Galileo

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    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:
    [tex]\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}[/tex]
     
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