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4角の中に当てはまる数字を教えてください

Let's fill in the numbers inside the boxes based on the provided mathematical functions and derivatives. The original equations and derivatives are given as:

\[ f(x) = \frac{6x^7 - 8}{9} \]
\[ f'(x) = -10x^{11} - 12x + 13 \]

The maximum area condition is achieved at:
\[ x = \frac{14}{15} \]
The maximum area is:
\[ f\left(\frac{14}{15}\right) = 16 \]

Given these, let’s fill in the boxes with the correct numbers:

1. **For \( f(x) \)**:

\[ f(x) = \frac{6x^7 - 8}{9} \]

The boxes should represent:
\[ f(x) = \frac{6x^7}{9} - \frac{8}{9} \]

Simplifying, we get:
\[ f(x) = \frac{2x^7}{3} - \frac{8}{9} \]

2. **For \( f'(x) \)**:

\[ f'(x) = -10x^{11} - 12x + 13 \]

Now, let’s rewrite \( f(x) \) and \( f'(x) \) with the numbers filled in:

### Corrected \( f(x) \):
\[ f(x) = \frac{2x^7}{3} - \frac{8}{9} \]

- The first box for \( f(x) \) should be \( 2 \)
- The second box for \( f(x) \) should be \( 3 \)
- The third box for \( f(x) \) should be \( 8 \)
- The fourth box for \( f(x) \) should be \( 9 \)

### Corrected \( f'(x) \):
\[ f'(x) = -10x^{11} - 12x + 13 \]

- The first box for \( f'(x) \) should be \( 10 \)
- The second box for \( f'(x) \) should be \( 11 \)
- The third box for \( f'(x) \) should be \( 12 \)
- The fourth box for \( f'(x) \) should be \( 13 \)

So the boxes should be filled as follows:

\[ f(x) = \frac{2x^7}{3} - \frac{8}{9} \]
\[ \frac{6}{9} x^7 - \frac{8}{9} = \frac{2}{3} x^7 - \frac{8}{9} \]

- \( f(x) \) boxes: \( 2, 3, 8, 9 \)
- \( f'(x) \) boxes: \( 10, 11, 12, 13 \)

Would you like further details or have any questions about this problem?

Here are 5 questions you might ask:
1. How do you derive a function like \( f(x) \)?
2. Can you explain the process of finding \( f'(x) \) in detail?
3. How can you verify the maximum value of a function?
4. What is the significance of critical points in optimization problems?
5. Can you explain how to solve for \( x \) when maximizing or minimizing a function?

**Tip:** When working with functions and their derivatives, always double-check your simplifications and algebraic manipulations to ensure accuracy.

Explore a maximization problem involving derivatives and function analysis. Learn how to find the maximum area using calculus and derivative techniques. Suitable for university-level students and anyone interested in advanced mathematical problem-solving.

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