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x values of local minimas and maximas, and inflection points, and where f(x) is concave up

To analyze the function \( f(x) = 11x - 4 \ln(x) \) with \( x > 0 \) and determine its critical points, points of inflection, and concavity, we proceed by finding the first and second derivatives of \( f(x) \) and analyzing their behavior.

### Step 1: Find the First Derivative \( f'(x) \)
To locate local minima and maxima, we first find the derivative and set it equal to zero.

\[
f'(x) = 11 - \frac{4}{x}
\]

Set \( f'(x) = 0 \):

\[
11 - \frac{4}{x} = 0
\]
\[
\frac{4}{x} = 11
\]
\[
x = \frac{4}{11}
\]

This value of \( x \) represents a critical point.

### Step 2: Determine the Nature of the Critical Point
To determine if this critical point is a minimum or maximum, we can use the second derivative test.

### Step 3: Find the Second Derivative \( f''(x) \)
The second derivative helps us analyze concavity and locate points of inflection.

\[
f''(x) = \frac{4}{x^2}
\]

Since \( f''(x) = \frac{4}{x^2} \) is always positive for \( x > 0 \), this means the function is always concave up on its entire domain \( (0, \infty) \).

### Summary of Results
- **Local Minima**: The critical point \( x = \frac{4}{11} \) is a minimum because the second derivative is positive.
- **No Local Maxima**: There are no points where \( f(x) \) has a local maximum.
- **Inflection Points**: There are no points of inflection because \( f''(x) \) does not change sign; it is always positive.
- **Concavity**: The function is concave up for all \( x > 0 \).

Would you like additional details on any part of this solution?

Here are some related questions you might consider:
1. What is the exact minimum value of \( f(x) \) at \( x = \frac{4}{11} \)?
2. How does the function \( f(x) \) behave as \( x \to \infty \)?
3. Can we sketch the graph of \( f(x) \) based on this information?
4. How does the presence of the \( \ln(x) \) term affect the behavior of \( f(x) \) near \( x = 0 \)?
5. What is the significance of concavity in interpreting the behavior of \( f(x) \)?

**Tip:** For analyzing concavity and inflection points, always check the sign changes in \( f''(x) \), as they indicate where the concavity might switch.

x values of local minima and maxima, and inflection points, and where f(x) is concave up.

Learn how to determine the critical points, inflection points, and concavity of the function f(x) = 11x - 4ln(x) with x > 0. This analysis involves using the first and second derivatives to identify local minima, maxima, and concave intervals. Suitable for advanced high school calculus students.

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