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

### Step 1: Find the Derivative of \( f(x) \)
To determine the critical points, let's differentiate \( f(x) \) with respect to \( x \):

\[
f(x) = \frac{25}{x} + 4x
\]

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

### Step 2: Set the Derivative to Zero
Now, set \( f'(x) = 0 \) to find the critical points:

\[
-\frac{25}{x^2} + 4 = 0
\]

Rearrange this equation:
\[
\frac{25}{x^2} = 4
\]

Multiply both sides by \( x^2 \):
\[
25 = 4x^2
\]

Divide by 4:
\[
x^2 = \frac{25}{4}
\]

Take the square root of both sides:
\[
x = \frac{5}{2} \quad \text{or} \quad x = -\frac{5}{2}
\]

Since \( x = -\frac{5}{2} \) is not in the interval \([1, 25]\), we discard it. So, the critical point within the interval is \( x = \frac{5}{2} \).

### Step 3: Evaluate \( f(x) \) at the Endpoints and Critical Point
Now, evaluate \( f(x) \) at \( x = 1 \), \( x = 25 \), and \( x = \frac{5}{2} \).

1. **At \( x = 1 \):**
   \[
   f(1) = \frac{25}{1} + 4 \times 1 = 25 + 4 = 29
   \]

2. **At \( x = 25 \):**
   \[
   f(25) = \frac{25}{25} + 4 \times 25 = 1 + 100 = 101
   \]

3. **At \( x = \frac{5}{2} \):**
   \[
   f\left(\frac{5}{2}\right) = \frac{25}{\frac{5}{2}} + 4 \times \frac{5}{2} = 10 + 10 = 20
   \]

### Step 4: Determine the Absolute Maximum
From the values we computed:

- \( f(1) = 29 \)
- \( f(25) = 101 \)
- \( f\left(\frac{5}{2}\right) = 20 \)

The highest value is \( f(25) = 101 \). Therefore, the absolute maximum value of \( f(x) \) on the interval \([1, 25]\) is:

\[
\boxed{101}
\]

Would you like more details on any specific step or have any questions?

### Related Questions
1. How do we determine if a critical point is a maximum or minimum within an interval?
2. Can absolute maximum and minimum values occur only at endpoints?
3. How would the problem change if the interval was not restricted to \([1, 25]\)?
4. What role does the derivative play in finding extreme values of a function?
5. How can the second derivative be used to confirm maximum or minimum points?

### Tip
Always check both endpoints and critical points within the interval when finding absolute maximum or minimum values.

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

Learn how to find the absolute maximum value of f(x) = 25/x + 4x on the interval [1, 25]. This solution involves calculus concepts like differentiation, critical points, and the Extreme Value Theorem. Suitable for advanced high school calculus students.

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