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The problem asks to determine the values of \( a \) and \( b \) such that the function \( f(x) = x^2 + ax + b \) has a minimum at the point \( (3, 9) \). Additionally, you are instructed to use the knowledge of the first derivative and second derivative to achieve the solution.

To solve this, let's break it down:

1. **First Condition (minimum at \( (3, 9) \)):**
   - The minimum occurs at \( x = 3 \), so the derivative of \( f(x) \), denoted \( f'(x) \), must be equal to zero at \( x = 3 \).
   - The value of the function at \( x = 3 \) should be 9, i.e., \( f(3) = 9 \).

2. **Second Condition (derivative analysis):**
   - Since the function has a minimum at \( x = 3 \), the second derivative \( f''(x) \) should be positive at that point, confirming it's a minimum.

### Step-by-step solution:
1. Start by finding the first derivative of \( f(x) \):

\[
f'(x) = 2x + a
\]

2. Set \( f'(3) = 0 \) to satisfy the minimum condition:

\[
2(3) + a = 0 \implies a = -6
\]

3. Now, find the value of \( b \). Since \( f(3) = 9 \), substitute \( x = 3 \) into \( f(x) \) and solve for \( b \):

\[
f(3) = 3^2 + (-6)(3) + b = 9
\]
\[
9 - 18 + b = 9 \implies b = 18
\]

Thus, \( a = -6 \) and \( b = 18 \).

### Final answer:
The values of \( a \) and \( b \) are:
- \( a = -6 \)
- \( b = 18 \)

Would you like any further details or clarifications on this problem? Here are some related questions you could explore:

1. How would the solution change if the function had a maximum at \( (3, 9) \) instead of a minimum?
2. What would the function look like graphically with these values for \( a \) and \( b \)?
3. How can you verify the result using the second derivative test?
4. What happens if the function is cubic instead of quadratic? How does that affect the conditions for a minimum?
5. How do you determine the critical points of a general quadratic function?

**Tip**: When dealing with quadratic functions, always start by checking the first and second derivatives to determine if the critical points are minima, maxima, or saddle points.

Determine the values of a and b so that the function f(x) = x^2 + ax + b has a minimum at (3, 9), showing full work and using the knowledge of Rate of Change (ROC).

Determine the values of a and b for the quadratic function f(x) = x^2 + ax + b to have a minimum at the point (3, 9). This problem uses key concepts such as derivatives, rate of change, and quadratic functions, and provides a detailed step-by-step solution. Suitable for Grades 10-12.

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