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The problem is asking you to apply the **Intermediate Value Theorem (IVT)** to the function \( f(x) = 2x + \tan\left(\frac{x}{5}\right) \) on the closed interval \([10, 15]\).

### **Intermediate Value Theorem (IVT)**:
The IVT states that if a function \( f(x) \) is continuous on a closed interval \([a, b]\), and if \( f(a) \) and \( f(b) \) take on opposite signs, then there exists at least one \( c \in (a, b) \) such that \( f(c) = 0 \) (or some value between \( f(a) \) and \( f(b) \)).

#### Steps:
1. **Check for continuity**:
   The function \( f(x) = 2x + \tan\left(\frac{x}{5}\right) \) consists of polynomial and trigonometric terms. Since both are continuous, the function is continuous on the closed interval \([10, 15]\).

2. **Evaluate the function at the endpoints of the interval**:
   We need to calculate \( f(10) \) and \( f(15) \):
   
   \[
   f(10) = 2(10) + \tan\left(\frac{10}{5}\right) = 20 + \tan(2)
   \]
   \[
   f(15) = 2(15) + \tan\left(\frac{15}{5}\right) = 30 + \tan(3)
   \]

   Using approximations for the tangent function:
   \[
   \tan(2) \approx -2.185
   \]
   \[
   \tan(3) \approx 0.1425
   \]

   So, we get:
   \[
   f(10) \approx 20 - 2.185 = 17.815
   \]
   \[
   f(15) \approx 30 + 0.1425 = 30.1425
   \]

   Since both \( f(10) \) and \( f(15) \) are positive, the Intermediate Value Theorem does not guarantee a solution for \( f(x) = 0 \), as there is no sign change between the endpoints.

3. **Possible solutions**:
   The question is asking which equation the IVT guarantees a solution for. Since the function values at both endpoints are between 5 and 15, the most likely answer is that the IVT guarantees a solution for \( f(x) = 15 \), since 15 lies within the range of the function values on the interval.

Thus, the correct answer is:

**Option D: \( f(x) = 15 \)**.

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Would you like further details or have any questions?

Here are 5 related questions:
1. What does the Intermediate Value Theorem tell us about a function's behavior?
2. How do we determine whether a function is continuous on a given interval?
3. Can you apply the IVT to a non-continuous function?
4. What happens if the function values at the endpoints are equal?
5. How can we approximate trigonometric values without a calculator?

**Tip**: The Intermediate Value Theorem requires that the function is continuous on a closed interval, and guarantees a solution for any value between the function's values at the interval's endpoints.

The Intermediate Value Theorem applied to f(x) = 2x + tan(x/5) on the closed interval [10, 15] guarantees a solution in [10, 15] to which of the following equations?

Discover how the Intermediate Value Theorem guarantees a solution for f(x) = 2x + tan(x/5) on the interval [10, 15]. Explore the function's continuity and how values like f(x) = 15 are determined. Suitable for advanced high school students or early university calculus courses.

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