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To find the critical numbers of the function \( f(x) = 5 - 10x + x^2 \), we need to follow these steps:

1. **Find the derivative of the function:**  
   The derivative, \( f'(x) \), gives us the slope of the tangent to the curve at any point.  
   \[
   f(x) = 5 - 10x + x^2
   \]
   Therefore, the derivative is:
   \[
   f'(x) = -10 + 2x
   \]

2. **Set the derivative equal to zero and solve for \( x \):**  
   Critical numbers occur where the derivative is zero or undefined. In this case:
   \[
   -10 + 2x = 0
   \]
   Solving for \( x \):
   \[
   2x = 10 \implies x = 5
   \]

3. **Conclusion:**  
   The only critical number for this function is \( x = 5 \).

The critical number is:
\[
5
\]

Would you like further details or have any questions?

Here are five related questions for further exploration:
1. How do you determine if a critical number is a maximum, minimum, or neither?
2. What is the second derivative test, and how is it used to classify critical points?
3. How would you find critical points for a function with higher powers, such as a cubic or quartic function?
4. What happens to the critical points if the function has discontinuities?
5. Can a function have critical points if its derivative does not exist at some points?

**Tip:** Always verify if the critical number falls within the domain of the original function to ensure it is valid.

Find the critical number(s), if any, of the function f(x) = 5 - 10x + x^2. (Give your answer in the form of a comma-separated list. Express numbers in exact form. Use symbolic notation and fractions where needed. Enter DNE if there are no critical numbers.)

Learn how to find the critical numbers of the quadratic function f(x) = 5 - 10x + x^2. This problem involves using calculus concepts like derivatives to identify critical points. Suitable for students in Grades 11-12.

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