The problem asks to compute the value of the integral $\int_2^4 x f''(x) \, dx$, where $f(x)$ is a differentiable function, and the graph of $f'(x)$ is provided.

Key insights:

• The integral involves $f''(x)$, which is the derivative of $f'(x)$. Thus, we can analyze the graph of $f'(x)$ to understand the behavior of $f''(x)$.
• The second derivative, $f''(x)$, corresponds to the slope of the graph of $f'(x)$.
• The given integral also has an extra factor of $x$ multiplying $f''(x)$.

Approach:

1. Step 1: Find $f''(x)$ from the graph of $f'(x)$:

   • The slope of the $f'(x)$ graph in the interval from $x = 2$ to $x = 4$ will give us $f''(x)$.

2. Step 2: Compute the integral:

   • Using the slopes from the graph of $f'(x)$, compute the integral $\int_2^4 x f''(x) \, dx$.

Let me examine the slopes from the graph and calculate this integral. I'll need to process this step-by-step.The value of the integral $\int_2^4 x f''(x) \, dx$ is $18$ in simplified form.

Would you like more details on the steps, or do you have any other questions?

Here are 5 related questions that could deepen your understanding:

1. What is the relationship between the graph of $f'(x)$ and the behavior of $f(x)$?
2. How do we interpret the second derivative $f''(x)$ from the graph of $f'(x)$?
3. What role does the multiplication by $x$ play in the integral?
4. Can the Fundamental Theorem of Calculus be applied to interpret this integral?
5. How would the result change if we altered the limits of integration?

Tip: When analyzing integrals that involve derivatives and slopes, it's crucial to relate geometric properties (like slopes and areas) to the function's behavior!