Identify the inflection points and local maxima and minima of the function graphed to the right. Identify the open intervals on which the function is differentiable and is concave up and concave down. Find the inflection point(s). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.

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To analyze the graph of a function for **inflection points**, **local maxima and minima**, **differentiable intervals**, and **concavity**, follow these systematic steps:

### 1. **Local Maxima and Minima**
- **Local Maximum**: A point where the function changes from increasing to decreasing. Visually, it's a peak.
- **Local Minimum**: A point where the function changes from decreasing to increasing. Visually, it's a trough.

**How to Identify:**
- Look for high points (peaks) and low points (troughs) on the graph.
- At a local maximum, the slope changes from positive to negative.
- At a local minimum, the slope changes from negative to positive.

### 2. **Inflection Points**
- **Inflection Point**: A point on the graph where the concavity changes (from concave up to concave down or vice versa).

**How to Identify:**
- Look for points where the graph changes from "cup-shaped" (concave up) to "cap-shaped" (concave down), or vice versa.
- At an inflection point, the second derivative changes sign.

### 3. **Concavity**
- **Concave Up**: The graph curves upwards (shaped like a cup, ∪).
- **Concave Down**: The graph curves downwards (shaped like a cap, ∩).

**How to Identify:**
- Observe the direction in which the graph is curving.
- Concave up regions indicate the function is accelerating upwards.
- Concave down regions indicate the function is decelerating or accelerating downwards.

### 4. **Differentiable Intervals**
- The function is **differentiable** where it is smooth and has no sharp corners, cusps, vertical tangents, or discontinuities.

**How to Identify:**
- Look for any breaks, sharp turns, or vertical tangents in the graph. These points are **not** differentiable.
- The **open intervals** between such points where the graph is smooth are the differentiable intervals.

### **Applying the Concepts:**

Suppose you have a graph (since I can't see it, I'll outline the general approach):

1. **Find Local Maxima and Minima**:
   - Identify all peaks and troughs.
   - Note their coordinates (e.g., Local maximum at (a, b), Local minimum at (c, d)).

2. **Determine Inflection Points**:
   - Look for points where the graph changes its curvature.
   - Confirm by checking where the concavity switches.
   - Note their coordinates if applicable (e.g., Inflection point at (e, f)).

3. **Identify Concave Up and Concave Down Intervals**:
   - Mark intervals where the graph is cup-shaped (concave up).
   - Mark intervals where the graph is cap-shaped (concave down).

4. **Establish Differentiable Intervals**:
   - Exclude points with sharp corners, cusps, vertical tangents, or discontinuities.
   - Define open intervals between these points where the function is smooth.

### **Example Answer Structure:**

- **Local Maximum**: (a, b)
- **Local Minimum**: (c, d)
- **Inflection Point(s)**: (e, f)
- **Differentiable Intervals**: $$(-\infty, p)$$, $$(p, q)$$, $$(q, \infty)$$ *(excluding points where the function is not differentiable)*
- **Concave Up Intervals**: $$(r, s)$$, $$(t, u)$$
- **Concave Down Intervals**: $$(v, w)$$, $$(x, y)$$

*Replace the placeholders (a, b, c, d, etc.) with the actual coordinates and interval bounds based on the specific graph.*

### **Final Note:**
Without the specific graph, I can't provide exact values or intervals. However, by following the above steps, you can systematically determine all required features of the function based on its graph.
To find the **inflection points**, **local maxima and minima**, **differentiable intervals**, and **concave up/down intervals** of a function, follow these steps: 1. **Local Maxima and Minima**: - Identify peaks and troughs on the graph. - Note where the slope changes from positive to negative (maxima) or negative to positive (minima). 2. **Inflection Points**: - Look for points where the graph changes from concave up to concave down or vice versa. 3. **Differentiable Intervals**: - Determine where the function is smooth without sharp corners or discontinuities. 4. **Concavity**: - Identify intervals where the graph is cup-shaped (concave up) or cap-shaped (concave down). **Example Answer Structure**: - **Local Maximum**: (a, b) - **Local Minimum**: (c, d) - **Inflection Point(s)**: (e, f) - **Differentiable Intervals**: $$(-\infty, p)$$, $$(p, q)$$, $$(q, \infty)$$ - **Concave Up Intervals**: $$(r, s)$$, $$(t, u)$$ - **Concave Down Intervals**: $$(v, w)$$, $$(x, y)$$ *Replace the placeholders with actual coordinates and intervals based on the specific graph.* **Final Note**: Without the graph, exact values can't be determined, but following these steps will help identify the required features.

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