The problem defines $g(x) = \int_0^x f(t) \, dt$, where $f(t)$ is the function shown in the graph. The task is to determine the interval on which $g(x)$ is decreasing, given the relationship between $g'(x)$ and $f(x)$:

• Key relationship: $g'(x) = f(x)$.
  $g(x)$ is decreasing wherever $g'(x) = f(x) < 0$, i.e., $f(t)$ is below the $t$-axis.

Analyzing the graph of $f(t)$:

1. From the graph:

   • $f(t) > 0$ for $0 < t < 1$ and $1 < t < 3$.
   • $f(t) < 0$ for $3 < t < 5$.
   • $f(t) = 0$ at $t = 3$ and $t = 5$.

2. Hence, $g(x)$ is decreasing for $3 < x < 5$, as $f(t) < 0$ in this interval.

Final Answer:

The interval is $3 < x < 5$.

Would you like a detailed breakdown of the integral or further clarification on the process? Here are five related questions to deepen your understanding:

1. How is the derivative $g'(x)$ connected to the behavior of $g(x)$?
2. What happens to $g(x)$ when $f(t) = 0$?
3. Can you find the value of $g(x)$ at specific points, like $g(3)$ or $g(5)$?
4. How does the sign of $f(t)$ influence the integral $g(x)$?
5. How can you verify the result using properties of integrals?

Tip:

When analyzing graphs of functions defined by integrals, focus on the derivative relationship $g'(x) = f(x)$ to understand increasing or decreasing behavior.