Determine the intervals for which the function shown is decreasing

To determine the intervals where the function shown in the graph is decreasing, you need to look at the sections of the graph where the function has a negative slope (i.e., where the graph is moving downwards as you move from left to right).

Here’s the process:

1. **Identify Critical Points**: Find where the function’s slope (first derivative) changes sign. These are usually the points where the graph reaches local maxima or minima.

2. **Observe the Slope**: The function is decreasing where the graph has a downward slope (negative derivative).

### Analysis of the Graph:
- The graph starts by increasing, then it reaches a peak (local maximum) before it starts decreasing.
- It decreases until it hits a local minimum, after which it increases again.

### Based on the visual analysis:
- The function decreases between the local maximum and local minimum.

### Determine Intervals:
- **First Interval**: From the local maximum to the local minimum.

Based on the general appearance of the graph:
- If \( x_1 \) is the x-coordinate of the local maximum and \( x_2 \) is the x-coordinate of the local minimum, the function is decreasing on the interval \((x_1, x_2)\).

You would need the exact values of \( x_1 \) and \( x_2 \) (which can be found if provided or estimated from the graph) to give a precise answer. If not provided, describe them as the points where the curve changes direction from increasing to decreasing and then back to increasing.

### Summary:
The function is decreasing on the interval \((x_1, x_2)\), where \( x_1 \) is the x-coordinate of the local maximum, and \( x_2 \) is the x-coordinate of the local minimum.

Would you like to explore further details or calculations regarding this problem?

### Follow-up Questions:
1. Do you have the exact x-values for the critical points?
2. Are you asked to calculate the exact intervals or just identify them visually?
3. Is there additional information provided about the function or its derivatives?
4. Would you like to discuss how to find these intervals algebraically?
5. Do you need help identifying local maxima and minima for similar problems?

### Tip:
When determining where a function is decreasing, always look for where the first derivative is negative, which corresponds to downward slopes on the graph.

Explore how to determine intervals where a function decreases using graphical analysis and derivative concepts. This detailed guide covers critical points, slope analysis, and graphical interpretation. Suitable for advanced high school students and anyone studying calculus.

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