https://file.aiquickdraw.com/mathbot/file/17266815606808msm2mu5.jpg

The image contains the following mathematical problem and steps:

### Given Function:
\[
f(x) = -3x^2 - 6x + 9
\]

### Critical Points:
The zeros of the function are found by solving:
\[
x = -3 \text{ or } x = 1
\]

A sign analysis is performed with critical points at \(x = -3\), \(x = -1\), and \(x = 1\). From the diagram:
- The derivative \(f'(x)\) changes from positive to negative, indicating a maximum at \(x = -1\).

### Maximum Value:
It states that the function reaches its maximum value when \(x = -1\). To find this value:
\[
f(-1) = 12
\]

### Volume Calculation:
A volume is calculated using the formula for the volume of a cone:
\[
V = \frac{1}{3} \pi r^2 h
\]
Here, \(r = 3\) and the height \(h = 12\), so:
\[
V = \frac{1}{3} \pi (3^3 \cdot 12) = \frac{39\pi}{3}
\]

### Radius Calculation:
Finally, the radius \(r\) is computed using the Pythagorean theorem:
\[
r = \sqrt{3^2 - 12^2}
\]

Let me know if you would like further details or clarification!

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Here are 5 related questions to consider:
1. How do you find the critical points of a quadratic function?
2. Why does the derivative sign change at a maximum point?
3. How is the formula for the volume of a cone derived?
4. Can you solve the equation \(r = \sqrt{3^2 - 12^2}\)?
5. What is the significance of the Pythagorean theorem in geometry?

**Tip**: Always double-check calculations when dealing with squares and square roots, as negative results under the square root imply complex numbers.

Find the maximum value of the quadratic function -3x^2 - 6x + 9, and calculate the volume and radius of a cone given specific dimensions.

Learn how to find the maximum value of the quadratic function -3x^2 - 6x + 9, perform derivative-based analysis, and calculate the volume and radius of a cone using key geometric formulas. Suitable for students in Grades 10-12.

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