https://r2.erweima.ai/mathbot/image/17267142961848felky4h.jpg

Let's analyze the polynomial graph from the image and solve the questions:

### a) Use interval notation to describe the intervals over which the graph is increasing.
Looking at the graph, it appears to be increasing in two intervals:
1. From the first local minimum to the local maximum.
2. From another local minimum to positive infinity.

The approximate intervals for increasing behavior seem to be:
- \( (-2, 0) \)
- \( (2, \infty) \)

Thus, the graph is increasing on the intervals \( (-2, 0) \cup (2, \infty) \).

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### b) How many local maximums exist for this graph?
From the graph, it seems there is **1 local maximum** around \( x = 0 \), where the curve reaches a peak.

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### c) What is the absolute minimum for this graph?
The absolute minimum seems to occur at the lowest point of the curve. Based on the graph, this appears to be around \( x = -3 \), with the corresponding \( y \)-value being approximately \( y = -15 \).

Thus, the absolute minimum is at the point \( (-3, -15) \).

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If you would like further details or have additional questions, feel free to ask!

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### Related Questions:
1. What are the intervals where the graph is decreasing?
2. How would you determine if a critical point is a local maximum or minimum?
3. Can you estimate the x-intercepts of this polynomial graph?
4. How would the second derivative help in analyzing this graph?
5. What happens to the graph as \( x \to \infty \) and \( x \to -\infty \)?

### Tip:
When analyzing a graph, always identify critical points (where the derivative is zero) to determine local extrema (maximums and minimums).

Consider the polynomial graph below in answering the following: (a) Use interval notation to describe the intervals over which the graph is increasing. (b) How many local maximums exist for this graph? (c) What is the absolute minimum for this graph?

Learn how to analyze a polynomial graph to find the intervals of increase, the number of local maxima, and the absolute minimum. This problem covers polynomial functions, interval notation, and critical points, suitable for students in Grades 11-12.

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