To find the minimum profit for the restaurant, we need to look for the critical points of the profit function [tex]\( p(x) = x^2 - 10x + 5000 \)[/tex]. Critical points can be found by taking the derivative of the function and setting it equal to zero.
1. Differentiate the profit function [tex]\( p(x) \)[/tex]:
[tex]\[
p'(x) = \frac{d}{dx}(x^2 - 10x + 5000)
\][/tex]
Using basic differentiation rules:
[tex]\[
p'(x) = 2x - 10
\][/tex]
2. Set the derivative equal to zero to find critical points:
[tex]\[
2x - 10 = 0
\][/tex]
Solve for [tex]\( x \)[/tex]:
[tex]\[
2x = 10
\][/tex]
[tex]\[
x = 5
\][/tex]
3. Substitute the critical point back into the profit function to find the corresponding profit:
[tex]\[
p(5) = (5)^2 - 10(5) + 5000
\][/tex]
Calculate the value:
[tex]\[
p(5) = 25 - 50 + 5000
\][/tex]
[tex]\[
p(5) = 4975
\][/tex]
Therefore, the minimum profit for the restaurant is \[tex]$4975.
The minimum profit for the restaurant is \$[/tex]__4975__.
