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To find the absolute maximum value of \( f(x) = \frac{25}{x} + 4x \) on the interval \([1, 25]\), we need to:

1. **Find the critical points** by taking the derivative of \( f(x) \) and setting it equal to zero.
2. **Evaluate** \( f(x) \) at the critical points and at the endpoints \( x = 1 \) and \( x = 25 \).
3. **Compare values** to find the maximum.

### Step 1: Differentiate \( f(x) \)
The function is:
\[
f(x) = \frac{25}{x} + 4x
\]
The derivative \( f'(x) \) is:
\[
f'(x) = -\frac{25}{x^2} + 4
\]

### Step 2: Set \( f'(x) = 0 \) to Find Critical Points
\[
-\frac{25}{x^2} + 4 = 0
\]
Solving for \( x \):
\[
\frac{25}{x^2} = 4
\]
\[
x^2 = \frac{25}{4}
\]
\[
x = \pm \frac{5}{2}
\]
Since \( x = \frac{5}{2} = 2.5 \) lies within the interval \([1, 25]\), it is a valid critical point.

### Step 3: Evaluate \( f(x) \) at \( x = 1 \), \( x = 25 \), and \( x = 2.5 \)
1. **At \( x = 1 \):**
   \[
   f(1) = \frac{25}{1} + 4 \cdot 1 = 25 + 4 = 29
   \]
2. **At \( x = 25 \):**
   \[
   f(25) = \frac{25}{25} + 4 \cdot 25 = 1 + 100 = 101
   \]
3. **At \( x = 2.5 \):**
   \[
   f(2.5) = \frac{25}{2.5} + 4 \cdot 2.5 = 10 + 10 = 20
   \]

### Step 4: Determine the Maximum Value
The values are:
- \( f(1) = 29 \)
- \( f(25) = 101 \)
- \( f(2.5) = 20 \)

Thus, the absolute maximum value on the interval \([1, 25]\) is **101** at \( x = 25 \).

Would you like further details on any step?

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### Related Questions:
1. How do we find the absolute minimum value of \( f(x) \) on the interval \([1, 25]\)?
2. What happens if we change the interval? How does the maximum value shift?
3. How would the procedure change if \( f(x) \) were given in a different form?
4. Can we use a graphical approach to verify the maximum?
5. What are other methods to find critical points of a function?

### Tip:
Always evaluate both critical points and endpoints when finding absolute extrema on a closed interval.

Find the absolute maximum value of f(x) = 25/x + 4x on the interval [1, 25].

Determine the absolute maximum value of the function f(x) = 25/x + 4x over the interval [1, 25] using calculus techniques. This problem involves finding critical points, evaluating endpoints, and applying the Extreme Value Theorem. Suitable for high school or introductory calculus students.

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