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.