The function f(x) has 3 properties: a) f(x) = 0 1 2 b) limx→0 f(x) x = 1 c) f(x + y) = f(x) + f(y) + x 2 y + xy2 . Use these properties and the h-definition to find f 0 (0)

We are tasked with finding the derivative \( f'(0) \) using the three properties of the function \( f(x) \):

1. \( f(0) = 0 \)
2. \( \lim_{x \to 0} \frac{f(x)}{x} = 1 \)
3. \( f(x + y) = f(x) + f(y) + x^2y + xy^2 \)

To find \( f'(0) \), we will use the limit definition of the derivative:

\[
f'(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}
\]

Since \( f(0) = 0 \), this simplifies to:

\[
f'(0) = \lim_{h \to 0} \frac{f(h)}{h}
\]

From property (2), we know:

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

Thus, using this result, we conclude:

\[
f'(0) = 1
\]

Therefore, the derivative of \( f(x) \) at \( x = 0 \) is:

\[
f'(0) = 1
\]

Would you like further details on any step or concept used? Here are 5 questions that may deepen your understanding:

1. How does the h-definition of a derivative work in general?
2. What does the property \( \lim_{x \to 0} \frac{f(x)}{x} = 1 \) imply about the function near \( x = 0 \)?
3. Why does the expression \( f(x + y) = f(x) + f(y) + x^2y + xy^2 \) help us analyze the structure of \( f(x) \)?
4. Can we infer anything about the higher-order derivatives of \( f(x) \) from this?
5. How can we generalize this process to find \( f'(a) \) for any \( a \)?

**Tip:** When solving limit-based problems, carefully examine how given conditions (like \( \lim_{x \to 0} \frac{f(x)}{x} = 1 \)) connect with definitions of derivative.

The function f(x) has 3 properties: a) f(x) = 0 1 2 b) limx→0 f(x) / x = 1 c) f(x + y) = f(x) + f(y) + x^2 y + xy^2. Use these properties and the h-definition to find f'(0).

Discover how to use the limit definition of derivatives and the given function properties to find f'(0). This problem explores key calculus concepts, including limits and derivative formulas. Ideal for undergraduate calculus students.

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