The profit function of the company is :
P(x) = -x² + 10x - 9
Here x is number of items produced each day.
Firstly, we will take the first derivative and equate it to 0 to find x :
P'(x) = d/dx(-x² + 10x - 9)
0 = -2x + 10
2x = 10
So, x = 5
Now, for maximum profit at x = 5, P"(x) should be negative.
Thus,
P"(x) = d/dx(-2x + 10)
So, P"(x) = -2 which represents maximum profit.
Therefore, for maximum profit, the number of items that must be produced is 5.