To solve this problem, we need to analyze the given functional equation:

$3f(x) + f(4 - x) = 6x^2 + 1.$

We are asked to find $f'(2) + f''(2)$.

Step 1: Find the Value of $f(x)$

The given equation can be considered for specific values of $x$. Let us choose $x = 2$ first:

$3f(2) + f(4 - 2) = 6 \cdot 2^2 + 1.$

This simplifies to:

$3f(2) + f(2) = 24 + 1 = 25.$

Combining like terms, we have:

$4f(2) = 25 \implies f(2) = \frac{25}{4}.$

Step 2: Differentiate the Functional Equation

Differentiate both sides of the equation $3f(x) + f(4 - x) = 6x^2 + 1$ with respect to $x$:

$3f'(x) - f'(4 - x) \cdot 1 = 12x.$

Now, substitute $x = 2$:

$3f'(2) - f'(4 - 2) = 12 \cdot 2 = 24.$

Since $f'(4 - 2) = f'(2)$, we have:

$3f'(2) - f'(2) = 24,$

$2f'(2) = 24 \implies f'(2) = 12.$

Step 3: Differentiate Again to Find $f''(2)$

Differentiate the equation $3f'(x) - f'(4 - x) = 12x$ again with respect to $x$:

$3f''(x) - (-f''(4 - x)) = 12.$

This simplifies to:

$3f''(x) + f''(4 - x) = 12.$

Substituting $x = 2$:

$3f''(2) + f''(4 - 2) = 12.$

Since $f''(4 - 2) = f''(2)$, we have:

$3f''(2) + f''(2) = 12,$

$4f''(2) = 12 \implies f''(2) = 3.$

Step 4: Find $f'(2) + f''(2)$

Now, we calculate $f'(2) + f''(2)$:

$f'(2) + f''(2) = 12 + 3 = 15.$

Conclusion

The value of $f'(2) + f''(2)$ is $\boxed{15}$.

Would you like more details or have any questions about the steps?

Here are 5 related questions to explore:

1. What is a functional equation, and how is it different from a standard algebraic equation?
2. How do you differentiate implicit functions involving different variables?
3. Can this approach be applied to other similar functional equations?
4. How does the symmetry in functions help solve equations involving $f(x)$ and $f(4 - x)$?
5. What are some common mistakes to avoid when differentiating functional equations?

Tip: Always check for symmetry or special properties in functional equations. This can simplify finding derivatives and evaluating functions at specific points.