Consider the function f ( x )

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 $f(x) = x^n + nx$.

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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.