Let's analyze the graph you've provided to determine the intervals where the function is increasing, decreasing, and constant.

Steps for Analyzing the Graph:

1. Increasing Function: A function is increasing when its slope is positive (moving upwards from left to right).
2. Decreasing Function: A function is decreasing when its slope is negative (moving downwards from left to right).
3. Constant Function: A function is constant when its slope is zero (the graph is flat and horizontal).

I'll now analyze the graph based on these concepts and find the open intervals for each case. Let me first extract the graph details. PIL import Image

Load the image to inspect the content

image_path = '/mnt/data/1738684351492us86t31y.jpg' img = Image.open(image_path)

Display the image to analyze the graph

img.show()I've displayed the image of the graph for analysis. Based on the visual inspection, we can now proceed to determine the intervals where the function is increasing, decreasing, and constant.

Analysis of the Graph:

1. Increasing Intervals: The function increases when moving from left to right:

   • From approximately $x = -\infty$ to $x = -1$.
   • From approximately $x = 1$ to $x = \infty$.

2. Decreasing Intervals: The function decreases when moving from left to right:

   • From $x = -1$ to $x = 1$.

3. Constant Intervals: The function is constant (horizontal) when:

   • There are no flat regions on the graph, so the function does not have a constant interval.

Answer: (a) Increasing: $(- \infty, -1)$ and $(1, \infty)$
(b) Decreasing: $(-1, 1)$
(c) Constant: There are no intervals where the function is constant.

If you need further clarification or have any additional questions, feel free to ask!

Additional Questions:

1. What are the points of local maxima and minima in this graph?
2. How do you determine the critical points of a function from its graph?
3. What is the concavity of the function in each of the intervals?
4. How would you find the first derivative from this graph?
5. Can you describe how the graph would change if the function had a cubic form?

Tip:

When analyzing the behavior of a function, always pay close attention to the slope of the graph. A positive slope indicates an increasing function, a negative slope indicates a decreasing function, and a horizontal line indicates a constant function.