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The image contains a mathematical expression written in Indonesian, asking for the minimum value of the following function for real numbers \( x \):

\[
2^x + 3^x + 4^x + 5^x + 2018 + 2^{-x} + 3^{-x} + 4^{-x} + 5^{-x}
\]

### Steps to solve this problem:

1. **Break the expression into two parts:**
   The given expression can be split into two groups:
   
   - \( f_1(x) = 2^x + 3^x + 4^x + 5^x \)
   - \( f_2(x) = 2^{-x} + 3^{-x} + 4^{-x} + 5^{-x} \)

   Thus, the total expression is:
   
   \[
   f(x) = f_1(x) + f_2(x) + 2018
   \]

2. **Symmetry analysis:**
   - The terms \( 2^x + 2^{-x}, 3^x + 3^{-x}, 4^x + 4^{-x}, 5^x + 5^{-x} \) are of the form \( a^x + a^{-x} \).
   - Each of these terms achieves a minimum when \( x = 0 \), as \( a^x + a^{-x} \geq 2 \) (by the AM-GM inequality).
   
3. **Evaluate at \( x = 0 \):**
   When \( x = 0 \), each term simplifies as follows:
   
   \[
   f(0) = 2^0 + 3^0 + 4^0 + 5^0 + 2018 + 2^0 + 3^0 + 4^0 + 5^0
   \]
   
   Since \( a^0 = 1 \) for any \( a \), this becomes:
   
   \[
   f(0) = 1 + 1 + 1 + 1 + 2018 + 1 + 1 + 1 + 1 = 2028
   \]

4. **Conclusion:**
   The minimum value of the function occurs at \( x = 0 \), and the value is \( 2028 \).

Would you like further details or clarifications?

### 5 Related Questions:
1. How do we prove that the function is minimized at \( x = 0 \)?
2. What are the applications of the AM-GM inequality in other types of problems?
3. How would this problem change if the constants were different, say \( 6^x \) or \( 7^x \)?
4. How can we solve similar minimization problems using derivatives?
5. What would happen if the constant \( 2018 \) were replaced by a variable \( c \)?

**Tip:** Always check if a function is symmetric or has certain properties like convexity when looking for its minimum or maximum values.

Untuk x bilangan real, berapakah nilai terkecil dari 2^x + 3^x + 4^x + 5^x + 2018 + 2^(-x) + 3^(-x) + 4^(-x) + 5^(-x)?

Learn how to find the minimum value of the function 2^x + 3^x + 4^x + 5^x + 2018 + 2^(-x) + 3^(-x) + 4^(-x) + 5^(-x) using symmetry analysis and the AM-GM inequality. This problem covers key mathematical concepts such as exponential functions and inequalities, suitable for students in Grades 10-12.

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