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.