Annual profit in thousands of dollars is given by the function,p(x)=200square x-3 +10,000,where x is the number of items sold in thousands, x less than 3.

describe the meaning of the numbers 3 and 10,000 in the formula.
find the profit for 5 different values of x.
graph the profit function over its given domain; use 5 values calculated in part 3 to construct the graph and connect these points with a smooth curve.
will this profit function have a maximum, if so what is it?

p(x)=-0.5x+40x-300 dollars,

how many items to be sold to get mazimun profit?

In the function, p(x) = 200(x^2 -3) + 10,000, the number 3 has a specific meaning. It represents the upper limit for the number of items sold in thousands. This means that the function is only defined for values of x that are strictly less than 3.

The number 10,000 in the formula represents the base profit or the profit when no items are sold. It is a constant term added to the product of 200 and (x^2 - 3) to calculate the total profit.

To find the profit for different values of x, we need to substitute those values into the function p(x) and then calculate the result. Let's calculate the profit for 5 different values of x.

For x = 1 (1,000 items sold):
p(1) = 200(1^2 - 3) + 10,000
= 200(1 - 3) + 10,000
= -400 + 10,000
= 9,600

For x = 2 (2,000 items sold):
p(2) = 200(2^2 - 3) + 10,000
= 200(4 - 3) + 10,000
= 200 + 10,000
= 10,200

For x = 2.5 (2,500 items sold):
p(2.5) = 200(2.5^2 - 3) + 10,000
= 200(6.25 - 3) + 10,000
= 200(3.25) + 10,000
= 650 + 10,000
= 10,650

For x = 2.7 (2,700 items sold):
p(2.7) = 200(2.7^2 - 3) + 10,000
= 200(7.29 - 3) + 10,000
= 200(4.29) + 10,000
= 858 + 10,000
= 10,858

For x = 2.9 (2,900 items sold):
p(2.9) = 200(2.9^2 - 3) + 10,000
= 200(8.41 - 3) + 10,000
= 200(5.41) + 10,000
= 1,082 + 10,000
= 11,082

To graph the profit function, we will plot the points (x, p(x)) with the values we calculated above on a coordinate plane. Connecting these points with a smooth curve will give us the graph of the profit function over its given domain.

Regarding the maximum profit, we can determine it by examining the graph. If the graph has a peak or the highest point, then the profit function will have a maximum.