Consider the function
f
(
x
)
=
x
n
+
n
x
f(x)=
n
x
​
+
x
n
​
from
(
0
,
∞
)
(0,∞) to
R
R, where
n
n is a positive integer.

Which of the following option(s) is(are) true?

n
n is a local maximum.

1
1 is a local maximum.

1
1 is a global maximum.

n
n is a local minimum.

n
n is a global minimum.

To analyze the properties of the function \( f(x) = x^n + nx \) on the interval \( (0, \infty) \) where \( n \) is a positive integer, we will examine the critical points and their nature (maximum or minimum) by calculating the first and second derivatives of \( f(x) \).

### 1. First Derivative
The first derivative of \( f(x) \) is:

\[
f'(x) = \frac{d}{dx}(x^n + nx) = nx^{n-1} + n
\]

### 2. Critical Points
To find the critical points, we set \( f'(x) = 0 \):

\[
nx^{n-1} + n = 0
\]

\[
n(x^{n-1} + 1) = 0
\]

Since \( n \) is positive, we simplify the equation to:

\[
x^{n-1} + 1 = 0
\]

However, \( x^{n-1} \) is non-negative for \( x > 0 \), thus there are **no real solutions** to \( x^{n-1} + 1 = 0 \). Therefore, there are no critical points from setting the first derivative to zero.

### 3. End Behavior and Nature of the Function
We examine the behavior of \( f(x) \) at the boundary of the interval \( (0, \infty) \):

- As \( x \to 0^+ \), \( f(x) \to 0 \) since \( x^n \to 0 \) and \( nx \to 0 \).
- As \( x \to \infty \), \( f(x) \to \infty \) since both \( x^n \) and \( nx \) grow to infinity.

### 4. Second Derivative Test
To determine the nature of the function (concave up or down), we compute the second derivative:

\[
f''(x) = \frac{d}{dx}(nx^{n-1} + n) = n(n-1)x^{n-2}
\]

Since \( n \) is positive:

- If \( n = 1 \), then \( f''(x) = 0 \), so the second derivative test is inconclusive, but \( f(x) = x + 1 \) is a line with a positive slope.
- If \( n > 1 \), then \( f''(x) > 0 \) for \( x > 0 \), indicating that \( f(x) \) is concave up, suggesting that there are no local maxima.

Given the absence of critical points and the concave up nature (or linear nature for \( n = 1 \)), we can conclude:

1. **\( n \) is not a local maximum**.
2. **1 is not a local maximum** as it is not a critical point.
3. **1 is not a global maximum**.
4. **\( n \) is not a local minimum** as it is not a critical point.
5. **\( n \) is not a global minimum** as \( f(x) \) increases to infinity as \( x \to \infty \).

### Conclusion
None of the provided statements are correct for the function \( f(x) = x^n + nx \).

---

Do you want details or have any questions? Here are some related questions for further exploration:

1. How can we determine if a function has a global minimum?
2. What does the second derivative test tell us about a function's concavity?
3. Why does the absence of critical points imply no local extrema for this function?
4. How does the value of \( n \) influence the shape of the function \( f(x) \)?
5. What are the implications of having no critical points within an interval for the function's extrema?
6. How would the function change if the domain included \( x = 0 \)?
7. What are some examples of functions that have both local and global extrema?
8. How does the linearity of the function for \( n = 1 \) affect its extrema?

**Tip:** Always consider the domain of a function when determining critical points and extrema. This ensures that all potential cases are covered.

Explore the function f(x) = x^n + nx on the interval (0, ∞) to determine the presence of critical points and extrema. Understand the behavior of the function, including its derivatives and the Second Derivative Test. Suitable for advanced high school students studying calculus.

Math Problem Statement

Consider the function f ( x )

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation