Question: Supposef: (0,+∞)→Ris a function such thatf′(1) =Aandfor any positive numbersx1, x2, we havef(x1x2) =x2f(x1)+x1f(x2). Findf(1) and prove thatf′(x0) =A+f(x0)x0for any pointx0>0. [Hint: Youmay need to use the substitutionx=yx0at some point.]

We are given that \( f : (0, +\infty) \to \mathbb{R} \) is a function satisfying:

1. \( f'(1) = A \) (where \( A \) is a constant),
2. For any \( x_1, x_2 > 0 \), the functional equation \( f(x_1 x_2) = x_2 f(x_1) + x_1 f(x_2) \) holds.

We are tasked with finding \( f(1) \) and proving that \( f'(x_0) = A + \frac{f(x_0)}{x_0} \) for any \( x_0 > 0 \).

### Step 1: Find \( f(1) \)
Let's set \( x_1 = x_2 = 1 \) in the functional equation \( f(x_1 x_2) = x_2 f(x_1) + x_1 f(x_2) \). This gives:
\[
f(1 \cdot 1) = 1 \cdot f(1) + 1 \cdot f(1),
\]
which simplifies to:
\[
f(1) = 2f(1).
\]
This equation implies that:
\[
f(1) = 0.
\]

### Step 2: Prove \( f'(x_0) = A + \frac{f(x_0)}{x_0} \)
We now aim to prove that \( f'(x_0) = A + \frac{f(x_0)}{x_0} \) for any \( x_0 > 0 \).

#### Substitution step:
Let \( x_1 = x_0 \) and \( x_2 = y \) in the functional equation \( f(x_1 x_2) = x_2 f(x_1) + x_1 f(x_2) \). This gives:
\[
f(x_0 y) = y f(x_0) + x_0 f(y).
\]
Now, differentiate both sides with respect to \( y \). Using the product rule on the right-hand side, we get:
\[
\frac{d}{dy} f(x_0 y) = f(x_0) + x_0 f'(y).
\]
On the left-hand side, using the chain rule, we have:
\[
f'(x_0 y) \cdot x_0 = f(x_0) + x_0 f'(y).
\]
Now, set \( y = 1 \) in this equation:
\[
f'(x_0) \cdot x_0 = f(x_0) + x_0 f'(1).
\]
Since \( f'(1) = A \), we substitute this into the equation:
\[
f'(x_0) \cdot x_0 = f(x_0) + x_0 A.
\]
Dividing through by \( x_0 \), we obtain:
\[
f'(x_0) = A + \frac{f(x_0)}{x_0}.
\]
This completes the proof.

### Final Answer:
- \( f(1) = 0 \),
- \( f'(x_0) = A + \frac{f(x_0)}{x_0} \) for any \( x_0 > 0 \).

Would you like more details or have any questions?

Here are 5 related questions you might find interesting:
1. What properties can be deduced about the function \( f(x) \) from the functional equation?
2. How does the condition \( f'(1) = A \) impact the behavior of \( f(x) \)?
3. What other techniques can be used to solve functional equations involving derivatives?
4. How does the result \( f(1) = 0 \) affect the form of the function \( f(x) \)?
5. Can we determine \( f(x) \) explicitly for any special cases?

**Tip**: When working with functional equations, try substituting simple values or introducing new variables to simplify the given equation.

Suppose f: (0,+∞)→R is a function such that f′(1) = A and for any positive numbers x1, x2, we have f(x1x2) = x2f(x1) + x1f(x2). Find f(1) and prove that f′(x0) = A + f(x0)/x0 for any point x0 > 0. [Hint: You may need to use the substitution x = yx0 at some point.]

Solve the functional equation f(x1x2) = x2f(x1) + x1f(x2), where f′(1) = A. Discover how to find f(1) and prove that f′(x0) = A + f(x0)/x0 for any x0 > 0 using differentiation techniques. This problem is suitable for undergraduate-level math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation