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Math Help - Challenge problem.

  1. #1
    Ted
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    Challenge problem.

    Hello.
    I have this problem, I stucked.
    I did not post it in the challenge sub-forum; because I do not know the solution.


    Problem: Suppose f is a function satisfies the equation f(x+y)=f(x)+f(y)+x^2y+xy^2 for all number x and y. Suppose also that: \lim_{x\to 0} \frac{f(x)}{x}=1.
    • (a) Find f(0).
    • (b) Find f'(0).
    • (c) Find f'(x).
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  2. #2
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    Quote Originally Posted by Ted View Post
    Hello.
    I have this problem, I stucked.
    I did not post it in the challenge sub-forum; because I do not know the solution.


    Problem: Suppose f is a function satisfies the equation f(x+y)=f(x)+f(y)+x^2y+xy^2 for all number x and y. Suppose also that: \lim_{x\to 0} \frac{f(x)}{x}=1.
    • (a) Find f(0).
    • (b) Find f'(0).
    • (c) Find f'(x).
    (a) f(0+0) = f(0) + f(0) + 0^2 \cdot 0 + 0 \cdot 0^2 <br />

    f(0) = 2f(0)

    f(0) = 0


    (b) f'(0) = \lim_{x \to 0} \frac{f(x) - f(0)}{x - 0}<br />

    f'(0) = \lim_{x \to 0} \frac{f(x)}{x} = 1


    (c) f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

    f'(x) = \lim_{h \to 0} \frac{f(x) + f(h) + x^2h + xh^2 - f(x)}{h}

    f'(x) = \lim_{h \to 0} \frac{f(h) + x^2h + xh^2}{h}<br />

    f'(x) = \lim_{h \to 0} \left[\frac{f(h)}{h}  + \frac{h(x^2 + xh)}{h}\right]<br />

    f'(x) = \lim_{h \to 0} \left[\frac{f(h)}{h}  + (x^2 + xh) \right]

    f'(x) = 1 + x^2
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