# Derivative Problem

• Aug 15th 2009, 02:18 PM
Derivative Problem
Suppose $\displaystyle f$ is a function that satisifies the equation $\displaystyle f(x+y)=f(x)+f(y)+yx^2+xy^2$ for all real $\displaystyle x,y$. Suppose that $\displaystyle \lim_{x->0}\frac{f(x)}{x}=1$. Find $\displaystyle f(0),f'(0),f'(x)$.

I star by writing $\displaystyle x+y=0,x=-y, y=-x$

$\displaystyle f(0)=f(x)+f(-x)+x^3-x^3$
$\displaystyle =f(x)+f(-x)$

So now I have:

$\displaystyle f(x)=f(0)-f(-x)$

$\displaystyle f(0)=f(0)-f(0)=0$

For $\displaystyle f'(0)$:

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

$\displaystyle =\lim_{x->0}\frac{f(x)}{x}=1$

So I got $\displaystyle f'(0)=1,f(0)=0$. I can't seem to get started on finding $\displaystyle f'(x)$.
• Aug 15th 2009, 02:36 PM
NonCommAlg
Quote:

Suppose $\displaystyle f$ is a function that satisifies the equation $\displaystyle f(x+y)=f(x)+f(y)+yx^2+xy^2$ for all real $\displaystyle x,y$. Suppose that $\displaystyle \lim_{x->0}\frac{f(x)}{x}=1$. Find $\displaystyle f(0),f'(0),f'(x)$.

I star by writing $\displaystyle x+y=0,x=-y, y=-x$

$\displaystyle f(0)=f(x)+f(-x)+x^3-x^3$
$\displaystyle =f(x)+f(-x)$

So now I have:

$\displaystyle f(x)=f(0)-f(-x)$

$\displaystyle f(0)=f(0)-f(0)=0$

For $\displaystyle f'(0)$:

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

$\displaystyle =\lim_{x->0}\frac{f(x)}{x}=1$

So I got $\displaystyle f'(0)=1,f(0)=0$. I can't seem to get started on finding $\displaystyle f'(x)$.

to get $\displaystyle f(0) = 0$ it's easier to put $\displaystyle y = 0.$ to find $\displaystyle f'(x)$: $\displaystyle f'(x)=\lim_{y \to 0} \frac{f(x+y)-f(x)}{y}=\lim_{y\to0} \left(\frac{f(y)}{y} + x^2 + xy \right)=x^2 + 1.$ this also gives you $\displaystyle f(x)=\frac{x^3}{3} + x.$
The text passage contains a mathematical problem involving a function and its derivative, along with a solution that partially answers the problem. The solution finds \( f(0) \) and \( f'(0) \), but the user is stuck on finding \( f'(x) \). The response provides the complete solution for \( f'(x) \) and \( f(x) \).
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Derivative Problem

• Aug 15th 2009, 02:18 PM
Derivative Problem
Suppose $\displaystyle f$ is a function that satisifies the equation $\displaystyle f(x+y)=f(x)+f(y)+yx^2+xy^2$ for all real $\displaystyle x,y$. Suppose that $\displaystyle \lim_{x->0}\frac{f(x)}{x}=1$. Find $\displaystyle f(0),f'(0),f'(x)$.

I star by writing $\displaystyle x+y=0,x=-y, y=-x$

$\displaystyle f(0)=f(x)+f(-x)+x^3-x^3$
$\displaystyle =f(x)+f(-x)$

So now I have:

$\displaystyle f(x)=f(0)-f(-x)$

$\displaystyle f(0)=f(0)-f(0)=0$

For $\displaystyle f'(0)$:

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

$\displaystyle =\lim_{x->0}\frac{f(x)}{x}=1$

So I got $\displaystyle f'(0)=1,f(0)=0$. I can't seem to get started on finding $\displaystyle f'(x)$.
• Aug 15th 2009, 02:36 PM
NonCommAlg
Quote:

Suppose $\displaystyle f$ is a function that satisifies the equation $\displaystyle f(x+y)=f(x)+f(y)+yx^2+xy^2$ for all real $\displaystyle x,y$. Suppose that $\displaystyle \lim_{x->0}\frac{f(x)}{x}=1$. Find $\displaystyle f(0),f'(0),f'(x)$.

I star by writing $\displaystyle x+y=0,x=-y, y=-x$

$\displaystyle f(0)=f(x)+f(-x)+x^3-x^3$
$\displaystyle =f(x)+f(-x)$

So now I have:

$\displaystyle f(x)=f(0)-f(-x)$

$\displaystyle f(0)=f(0)-f(0)=0$

For $\displaystyle f'(0)$:

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

$\displaystyle =\lim_{x->0}\frac{f(x)}{x}=1$

So I got $\displaystyle f'(0)=1,f(0)=0$. I can't seem to get started on finding $\displaystyle f'(x)$.

to get $\displaystyle f(0) = 0$ it's easier to put $\displaystyle y = 0.$ to find $\displaystyle f'(x)$: $\displaystyle f'(x)=\lim_{y \to 0} \frac{f(x+y)-f(x)}{y}=\lim_{y\to0} \left(\frac{f(y)}{y} + x^2 + xy \right)=x^2 + 1.$ this also gives you $\displaystyle f(x)=\frac{x^3}{3} + x.$
user
The text passage contains a mathematical problem involving a function and its derivative, along with a solution that partially answers the problem. The solution finds \( f(0) \) and \( f'(0) \), but the user is stuck on finding \( f'(x) \). The response provides the complete solution for \( f'(x) \) and \( f(x) \).

TYPE: SINGLE_QUESTION_WITH_ANSWER