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The question is asking for the critical number of the function \( -9x^2 + 18x + 6 \).

To find the critical number, we first need to calculate the derivative of the function and set it equal to zero:

1. The function is \( f(x) = -9x^2 + 18x + 6 \).
2. Find the derivative: 
   \[
   f'(x) = -18x + 18
   \]
3. Set the derivative equal to zero to find the critical number:
   \[
   -18x + 18 = 0
   \]
   Solving for \(x\), we get:
   \[
   -18x = -18 \quad \Rightarrow \quad x = 1
   \]

Thus, the critical number is \( x = 1 \).

Would you like further details or have any other questions?

Here are five related questions to explore:
1. What are critical points and how do they relate to extrema?
2. How can you determine if the critical point is a maximum, minimum, or neither?
3. How do you find critical numbers for polynomial functions of higher degrees?
4. What role does the second derivative play in analyzing critical points?
5. How does the nature of critical points change in non-polynomial functions?

**Tip:** Always double-check if a critical point corresponds to an actual maximum or minimum by using the second derivative or first derivative test.

Find the critical number of the function -9x^2 + 18x + 6.

Learn how to find the critical number for the quadratic function -9x^2 + 18x + 6 by calculating its derivative and solving for where it equals zero. This problem is ideal for high school calculus students in Grades 10-12.

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