Suppose a function f x( ) satisfies the following conditions:

• f a b f a f b ab ( ) () () 2 += + + for all numbers a and b

• 0

( ) lim 7

h

f h

→ h =

Use the definition of the derivative to find f x ′( ).

We are given that a function \( f(x) \) satisfies two conditions:

1. \( f(a + b) = f(a) + f(b) + ab \) for all real numbers \( a \) and \( b \).
2. \( \lim_{h \to 0} \frac{f(h)}{h} = 0 \).

We are asked to find \( f'(x) \) using the definition of the derivative. Let's proceed step-by-step.

### Step 1: Use the derivative definition
The derivative of \( f(x) \) at a point \( x \) is defined as:

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

To apply this, we first need to evaluate \( f(x + h) - f(x) \). Using the given functional equation:

\[
f(x + h) = f(x) + f(h) + xh.
\]

Thus, we can express \( f(x + h) - f(x) \) as:

\[
f(x + h) - f(x) = (f(x) + f(h) + xh) - f(x) = f(h) + xh.
\]

### Step 2: Substitute into the derivative formula
Now we substitute this expression for \( f(x + h) - f(x) \) into the derivative formula:

\[
f'(x) = \lim_{h \to 0} \frac{f(h) + xh}{h}.
\]

We can separate the terms in the numerator:

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

### Step 3: Use the given limit
We are given that \( \lim_{h \to 0} \frac{f(h)}{h} = 0 \). So, applying this in the above expression:

\[
f'(x) = 0 + x = x.
\]

### Final Answer:
Thus, the derivative of \( f(x) \) is:

\[
f'(x) = x.
\]

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Would you like further details on this solution, or do you have any questions?

Here are five related questions for further exploration:

1. How does the functional equation \( f(a + b) = f(a) + f(b) + ab \) lead to \( f(x) = \frac{x^2}{2} \) as a possible solution?
2. How can we verify that the function \( f(x) = \frac{x^2}{2} \) satisfies the given functional equation?
3. What is the significance of the limit condition \( \lim_{h \to 0} \frac{f(h)}{h} = 0 \)?
4. How would the derivative change if the functional equation was \( f(a + b) = f(a) + f(b) \) instead?
5. Can we generalize the solution to find derivatives for other similar functions?

**Tip:** When using the derivative definition, always look for ways to manipulate the functional equation into a form where you can apply the given limits or other properties.

Suppose a function f(x) satisfies the following conditions: • f(a + b) = f(a) + f(b) + ab for all numbers a and b • lim (h -> 0) f(h) / h = 0. Use the definition of the derivative to find f'(x).

This problem involves finding the derivative of a function f(x) using functional equations and the definition of the derivative. Learn how to apply key concepts from calculus, such as limits and functional relationships, to solve this advanced problem.

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