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To find the absolute extrema (both the minimum and maximum) of the function \( y = \tan\left(\frac{\pi x}{28}\right) \) on the closed interval \([0, 7]\), we can follow these steps:

### 1. **Identify the function**:
   The function given is \( y = \tan\left(\frac{\pi x}{28}\right) \).

### 2. **Find the derivative**:
   The derivative of the function will help us locate any critical points in the interval where extrema might occur.
   
   The derivative of \( y = \tan\left(\frac{\pi x}{28}\right) \) is:
   \[
   y' = \frac{d}{dx} \left[\tan\left(\frac{\pi x}{28}\right)\right]
   = \sec^2\left(\frac{\pi x}{28}\right) \cdot \frac{\pi}{28}
   \]
   So, \( y' = \frac{\pi}{28} \sec^2\left(\frac{\pi x}{28}\right) \).

### 3. **Critical points**:
   Critical points occur where the derivative is zero or undefined. However, the derivative \( \sec^2\left(\frac{\pi x}{28}\right) \) is never zero, but it could become undefined when the tangent function itself is undefined (at its vertical asymptotes). The tangent function is undefined where:
   \[
   \frac{\pi x}{28} = \frac{\pi}{2} + n\pi, \quad n \in \mathbb{Z}
   \]
   Solving for \( x \):
   \[
   x = 14n + 7
   \]
   The smallest such value of \( x \) in the interval \([0, 7]\) is at \( x = 7 \), so there is a vertical asymptote at \( x = 7 \).

### 4. **Check values at endpoints**:
   Now, we check the function at the boundaries of the interval \( [0, 7] \).

   - At \( x = 0 \):
     \[
     y(0) = \tan\left(\frac{\pi \cdot 0}{28}\right) = \tan(0) = 0
     \]
   - At \( x = 7 \), the function tends toward infinity (since it is near a vertical asymptote):
     \[
     y(7) = \tan\left(\frac{\pi \cdot 7}{28}\right) = \tan\left(\frac{\pi}{4}\right) = 1
     \]

### 5. **Conclusion**:
   The function reaches its **minimum** at \( x = 0 \), where \( y(0) = 0 \), and its **maximum** at \( x = 7 \), where \( y(7) = 1 \).

Thus, the absolute extrema are:
- Minimum: \( (0, 0) \)
- Maximum: \( (7, 1) \)

Would you like any further details or explanations on this process? Let me know!

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Here are 5 related questions to expand your understanding:
1. How do vertical asymptotes affect the behavior of functions like tangent?
2. Can you find the critical points for other trigonometric functions, such as \( \sin(x) \) or \( \cos(x) \)?
3. How would you compute the second derivative to check the concavity of the tangent function?
4. What is the importance of checking the function at the endpoints in closed interval problems?
5. How would you graph the function \( y = \tan\left(\frac{\pi x}{28}\right) \) on the interval \( [0, 7] \)?

**Tip**: When solving for extrema on closed intervals, always remember to check both the critical points and the function values at the boundaries.

Find the absolute extrema of the function y = tan(πx / 28) on the closed interval [0, 7].

Learn how to find the absolute minimum and maximum of the function y = tan(πx / 28) on the closed interval [0, 7]. This problem includes key trigonometric concepts, derivative calculations, and involves understanding critical points and endpoints.

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