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Annual profit for a restaurant is modeled by the function [tex][tex]$p = x^2 - 10x + 5000$[/tex][/tex].

Find the minimum profit for the restaurant. Enter your answer in dollars without commas or decimals.

The minimum profit for the restaurant is [tex]\$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_[/tex].



Answer :

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__.