The problem asks to evaluate the integral

$\int_0^5 |f'(x)||f(x)|^2 \, dx$

based on the graph of $y = f(x)$.

Step-by-step solution:

1. Interpret the graph:

   • $f(x)$ crosses the $x$-axis at $x = 1$ and $x = 4$.
   • $f(x) > 0$ in $(0, 1)$ and $(4, 5)$.
   • $f(x) < 0$ in $(1, 4)$.

2. Behavior of $f'(x)$:

   • $f'(x) > 0$ in $(0, 3)$ since $f(x)$ is increasing.
   • $f'(x) < 0$ in $(3, 5)$ since $f(x)$ is decreasing.

3. Expression for the integral: Since the absolute values of $f'(x)$ and $f(x)$ are given, we only work with the magnitudes. Breaking the integral into intervals based on $f(x)$'s zeros: $\int_0^5 |f'(x)||f(x)|^2 \, dx = \int_0^1 f'(x)f(x)^2 \, dx + \int_1^4 -f'(x)f(x)^2 \, dx + \int_4^5 f'(x)f(x)^2 \, dx.$

4. Evaluate $f(x)^2$ in each interval: From the graph:

   • $f(x)^2 = 3^2 = 9$ at $x = 0$,
   • $f(x)^2 = 0$ at $x = 1$, $x = 4$,
   • $f(x)^2 = 3^2 = 9$ again near $x = 5$.

   The graph suggests symmetry in the rise and fall of $f(x)$, implying symmetry in contributions to the integral.

5. Compute contributions:

   • The contribution of the integral in the positive intervals ($0 \to 1$, $4 \to 5$) cancels the contribution of the negative interval ($1 \to 4$).
   • Approximate contributions show that the result balances and evaluates to 10.

Thus, the answer is (e) 10.

Do you want detailed explanations on any step?

Related Questions:

1. How do we compute integrals involving absolute values directly from graphs?
2. How does symmetry in graphs help simplify integrals?
3. Why does the derivative $f'(x)$ affect the integral more in non-linear cases?
4. What are the typical uses of absolute value integrals in real-world problems?
5. Can numerical integration approximate this result if we lack analytical symmetry?

Tip:

Always examine critical points and zeros of a function when dealing with absolute value integrals, as they guide partitioning and simplification.