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Let me extract and interpret the math problems from the image, then I'll work through the solutions for you.The image contains the following mathematical exercise:

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**Given the function \( f(x) = x^3 - 6x^2 + 9x + 1 \):**

1. **Determine the value of the second derivative, \( f''(x) \), and use it to find intervals where the graph of \( f \) is concave upward or concave downward.**
2. **Find any points of inflection for the function.**
3. **Sketch the graph of \( f(x) \).**

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Let’s start with solving the problem step by step:

### Step 1: Finding the Second Derivative

The first step is to find the first derivative \( f'(x) \) and then the second derivative \( f''(x) \).

1. \( f(x) = x^3 - 6x^2 + 9x + 1 \)
2. First derivative \( f'(x) \):
   \[
   f'(x) = 3x^2 - 12x + 9
   \]
3. Second derivative \( f''(x) \):
   \[
   f''(x) = 6x - 12
   \]

### Step 2: Concavity

To

Determine where the graph of the function is concave upward or downward, find any inflection points, and sketch the graph of the function f(x) = x^3 - 6x^2 + 9x + 1.

Analyze and sketch the graph of f(x) = x^3 - 6x^2 + 9x + 1 by determining the intervals of concavity and identifying any points of inflection. This solution involves key calculus concepts such as differentiation, concavity, and the Second Derivative Test, making it suitable for Grades 11-12.

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