determine absolute maximum (exact answer, no decimal) of In(x+72) + 1/x on [-71, -1], give answer as unsimplified

To find the absolute maximum of the function \( f(x) = \ln(x + 72) + \frac{1}{x} \) on the interval \( [-71, -1] \), we'll proceed by evaluating the endpoints and critical points.

### Step-by-Step Solution

1. **Find the derivative of \( f(x) \):**
   \[
   f(x) = \ln(x + 72) + \frac{1}{x}
   \]
   Differentiating with respect to \( x \):
   \[
   f'(x) = \frac{1}{x + 72} - \frac{1}{x^2}
   \]

2. **Set \( f'(x) = 0 \) to find critical points:**
   \[
   \frac{1}{x + 72} - \frac{1}{x^2} = 0
   \]
   This simplifies to:
   \[
   \frac{x^2 - (x + 72)}{x^2(x + 72)} = 0
   \]
   which further simplifies to:
   \[
   x^2 - x - 72 = 0
   \]
   Solving this quadratic equation using the quadratic formula:
   \[
   x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-72)}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 + 288}}{2} = \frac{1 \pm \sqrt{289}}{2}
   \]
   \[
   x = \frac{1 \pm 17}{2}
   \]
   Thus, \( x = 9 \) or \( x = -8 \).

   Since we are restricted to the interval \( [-71, -1] \), only \( x = -8 \) lies within this interval.

3. **Evaluate \( f(x) \) at the endpoints and at \( x = -8 \):**
   - **At \( x = -71 \):**
     \[
     f(-71) = \ln(-71 + 72) + \frac{1}{-71} = \ln(1) - \frac{1}{71} = 0 - \frac{1}{71} = -\frac{1}{71}
     \]

   - **At \( x = -1 \):**
     \[
     f(-1) = \ln(-1 + 72) + \frac{1}{-1} = \ln(71) - 1
     \]

   - **At \( x = -8 \):**
     \[
     f(-8) = \ln(-8 + 72) + \frac{1}{-8} = \ln(64) - \frac{1}{8} = \ln(64) - \frac{1}{8}
     \]

4. **Compare values:**
   - \( f(-71) = -\frac{1}{71} \)
   - \( f(-1) = \ln(71) - 1 \)
   - \( f(-8) = \ln(64) - \frac{1}{8} \)

   Since \( \ln(64) = 4 \ln(2) \), and \( \ln(71) > 4 \ln(2) \), the largest value is \( f(-1) = \ln(71) - 1 \).

### Conclusion
The absolute maximum of \( f(x) \) on \( [-71, -1] \) is:
\[
\boxed{\ln(71) - 1}
\]

Do you have any questions or want more details?

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### Related Questions

1. How do you find the absolute minimum of a function over a closed interval?
2. What are the conditions for a function to achieve absolute extrema over a closed interval?
3. How does the natural logarithm function \( \ln(x) \) behave on the interval \( (0, \infty) \)?
4. Why is it necessary to check endpoints when finding absolute extrema on a closed interval?
5. How can the quadratic formula be used to find critical points for more complex functions?

**Tip:** Always verify if critical points lie within the interval before evaluating the function at these points.

Determine the absolute maximum (exact answer, no decimal) of ln(x+72) + 1/x on [-71, -1], give answer as unsimplified.

Find the exact absolute maximum of ln(x+72) + 1/x on the interval [-71, -1] with a step-by-step solution. Explore calculus concepts, logarithmic functions, and critical points evaluation, suitable for high school calculus students.

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