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The question is asking to find critical points for the function:
\[
y = f(x) = 3x - \ln(4x)
\]
on the interval \(1 \leq x \leq 2.7\). You are to determine the global minimum, global maximum, and a local non-global maximum, rounding your answers to two decimal places.

### Solution Approach

1. **Find the derivative \( f'(x) \):**
   \[
   f(x) = 3x - \ln(4x)
   \]
   Differentiate with respect to \( x \):
   \[
   f'(x) = 3 - \frac{1}{x}
   \]

2. **Set \( f'(x) = 0 \) to find critical points:**
   Solve \( 3 - \frac{1}{x} = 0 \):
   \[
   3 = \frac{1}{x} \implies x = \frac{1}{3}
   \]
   This value is outside the given interval \([1, 2.7]\), so we do not consider it as a critical point within this range.

3. **Evaluate \( f(x) \) at the endpoints:**
   - **At \( x = 1 \):**
     \[
     f(1) = 3 \cdot 1 - \ln(4 \cdot 1) = 3 - \ln(4) \approx 3 - 1.39 = 1.61
     \]
   - **At \( x = 2.7 \):**
     \[
     f(2.7) = 3 \cdot 2.7 - \ln(4 \cdot 2.7) = 8.1 - \ln(10.8) \approx 8.1 - 1.475 = 6.625
     \]

4. **Identify the global minimum, maximum, and any local non-global maxima:**
   - **Global Minimum:** \( (x, y) = (1, 1.61) \)
   - **Global Maximum:** \( (x, y) = (2.7, 6.625) \)

   Since \(x = 1\) and \(x = 2.7\) are the only points in the interval where \(f(x)\) changes, they represent the required global maximum and minimum.

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Would you like to see more detailed steps or clarifications? Here are five related questions that might help you understand this type of problem better:

1. How do you identify global extrema on a closed interval?
2. What are the differences between local and global extrema?
3. How can we interpret the behavior of a logarithmic function in extrema problems?
4. How do we handle critical points that fall outside the interval of interest?
5. How do you calculate natural logarithms when estimating values?

**Tip:** Always check the endpoints of an interval when finding global extrema, as they might provide the highest or lowest values for the function over the interval.

Find the points at which y = f(x) = 3x - ln(4x) has a global maximum, a global minimum, and a local, non-global maximum on the interval 1 ≤ x ≤ 2.7. Round your answers to two decimal places.

Explore how to find the global maximum, global minimum, and local maximum of the function y = f(x) = 3x - ln(4x) on the interval [1, 2.7]. This calculus problem involves differentiation and optimization techniques.

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