https://r2.erweima.ai/mathbot/image/1726587165864rqes7yz9.jpg

We are given the sequence of functions \( f_n(x) = \frac{x}{1 + n x} \) for \( x \in [0, 1] \). The problem asks for the \( \|f_n\|_\infty \), which represents the supremum norm (or the maximum value) of the function \( f_n(x) \) on the interval \( [0, 1] \).

To compute \( \|f_n\|_\infty \), we need to find the maximum value of \( f_n(x) \) over the interval \( [0, 1] \) for each \( n \). Let's go through the steps:

### Step 1: Analyze the function \( f_n(x) = \frac{x}{1 + n x} \)
- The function is continuous on the closed interval \( [0, 1] \), so it attains its maximum on this interval.

### Step 2: Find the critical points
To find the maximum, we compute the derivative of \( f_n(x) \) and set it to zero:
\[
f_n'(x) = \frac{d}{dx} \left( \frac{x}{1 + n x} \right)
\]
Using the quotient rule:
\[
f_n'(x) = \frac{(1 + n x) \cdot 1 - x \cdot n}{(1 + n x)^2} = \frac{1 + n x - n x}{(1 + n x)^2} = \frac{1}{(1 + n x)^2}
\]
This derivative is always positive for \( x \in [0, 1] \), meaning that the function is increasing on the interval.

### Step 3: Evaluate \( f_n(x) \) at the endpoints
Since the function is increasing, the maximum will occur at \( x = 1 \). So, we evaluate \( f_n(1) \):
\[
f_n(1) = \frac{1}{1 + n}
\]

### Conclusion:
The supremum norm \( \|f_n\|_\infty \) is the maximum value of \( f_n(x) \), which occurs at \( x = 1 \). Therefore,
\[
\|f_n\|_\infty = \frac{1}{1 + n}
\]

Would you like more details on any of these steps or explanations? Here are five related questions you might find useful:

1. What does the supremum norm represent in the context of functions?
2. How would the behavior of \( f_n(x) \) change as \( n \to \infty \)?
3. Can you find the limit of \( f_n(x) \) as \( n \) grows large for a fixed \( x \in [0, 1] \)?
4. How does the quotient rule help when differentiating rational functions?
5. What are the general steps to analyze the maximum or minimum of a continuous function on a closed interval?

**Tip:** When analyzing functions for extremum values on closed intervals, always check both the critical points and endpoints!

Soit f_n(x) = \frac{x}{1 + n x} pour x \in [0, 1]. Quelle est \|f_n\|_\infty pour cette suite?

Explore the solution to the supremum norm of the function f_n(x) = x / (1 + nx) on the interval [0, 1]. Learn how to compute critical points using the quotient rule and find the maximum value of the function as n approaches infinity. Suitable for undergraduate students studying calculus.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation